On the parabolic kernel of the Schr6dinger operator
by
PETER LI(l) and SHING TUNG YAU
University of Utah Salt Lake City, UT, U.S.A.
University of California, San Diego La Jolla, CA, U.S.A.
Table of contents
w0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w1. Gradient estimates . . . . . . . . . . . . . . . . . . . . . . . . . w2. Harnack inequalities . . . . . . . . . . . . . . . . . . . . . . . . w3. Upper bounds of fundamental solutions . . . . . . . . . . . . . . w4. Lower bounds of fundamental solutions . . . . . . . . . . . . . . w5. Heat equation and Green's kernel . . . . . . . . . . . . . . . . . w6. The Schr6dinger operator . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 155 166 170 181 190 196 199 200
w0. Introduction
In this paper, we will study parabolic equations of the type
(Aq(x,
t)~)u(x,
t)=O
(0.1)
on a general Riemannian manifold. The function q(x, t) is a s s u m e d to be C 2 in the first variable and C 1 in the second variable. In classical situations [20], a H a r n a c k inequality for positive solutions was established locally. However, the geometric d e p e n d e n c y of the estimates is c o m p l i c a t e d and s o m e t i m e s unclear. Our goal is to prove a H a r n a c k inequality for p o s i t i v e solutions o f (0.1) (w 2) b y utilizing a gradient estimate derived in w 1. The m e t h o d o f p r o o f is originated in [26] and [8], w h e r e they have studied the elliptic case, i.e. the solution is time independent. In s o m e situations ( T h e o r e m s 2.2 and (1) Research partially supported by a Sloan fellowship and an NSF grant. 11868283 Acta Mathematica 156. Imprim6le 15 mai 1986
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PETER LI AND SHINGTUNGYAU
2.3), the Harnack inequality is valid globally, which enables us to relate the global geometry with the analysis. In w3, we apply the Harnack inequality to obtain upper estimates for the fundamental solution of the equation ( A  q ( x )  ~ t t ) u(x, t)=0,
(0.2)
where q is a function on M alone. We shall point out that for the heat equation (q=0), upper estimates for the heat kernel were obtained in [7] and [5]. However the estimate which we obtain is so far the sharpest, especially for large time. When the Ricci curvature is nonnegative the sharpness is apparent, since a comparable lower bound is also obtained in w4. A lower bound for the fundamental solution of (0.2) is also derived for some special situations. Applications of these estimates for the heat kernel are discussed in w5. A generalization of Widder's [25] uniqueness theorem for positive solutions of the heat equation is proved (2) (Theorem 5.1). In fact, the condition on the curvature is best possible due to the counterexample of Azencott [2]. We also point out that generalizations of Widder's theorem to general elliptic operators in R n were derived in [21], [1 l] and [l]. When M has nonnegative Ricci curvature, sharp upper and lower bounds of Green's function are derived. This can also be viewed as a necessary and sufficient condition for the existence of Green's function which was studied in [23]. In fact, in [24], our estimates on Green's function were proved for nonnegatively Ricci curved manifolds with pole and with nonnegative radial sectional curvatures. In [13], Gromov proved lower bounds for all the eigenvalues of the Laplacian on a compact nonnegatively Ricci curved manifold without boundary. We generalized these estimates to allow the manifold to have convex boundaries with either Dirichlet or Neumann boundary conditions. These lower bounds can also be viewed as a generalization of the lower bound for the first eigenvalue obtained in [16]. Another application is to derive an upper bound of the first Betti number, bi, on a compact manifold in terms of its dimension, a lower bound of the Ricci curvature, and an upper bound of the diameter. The manifold is allowed to have convex boundaries, in which case bl can be taken to be the dimension of either HI(M) of HI(M, aM). It was proved in [14] that if M has no boundary, then bk can be estimated from above by the
(2) Duringthe preparation of this paper, H. Donnellyhas independentlyfounda differentproofof the uniqueness theorem when the Ricci curvature of M is boundedfrom below.
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
155
dimension, k, a lower bound of the sectional curvature, and an upper bound of the diameter.(3) On bk, for k > l , we derived a weaker estimate than that in [14] assuming both upper bound of the sectional curvature and lower bound of the Ricci curvature. However, in this case, the manifold is also allowed to have nonempty convex boundaries. Some of our estimates on the Betti numbers overlap with results in [17], [18], and [19]. Finally, in w6, we study the asymptotic behaviour for the fundamental solution of the operator A22q(x)a/at as ;t,oo. This formula was needed in [22] for the understanding of multiplewelled potentials. In fact, the results in [22] can be carded over to any complete manifold after applying the formula in Theorem 6.1.
w1. Gradient estimates Throughout this section, M is assumed to be an ndimensional complete Riemannian manifold with (possibly empty) boundary, aM. Let O/av be the outward pointing unit normal vector to aM, and denote the second fundamental form of aM with respect to
a/av by II.
Our goal is to derive estimates on the derivates of positive solutions u(x, t) on M x (0, 0o) of the equation
(Aq(x, t)~)u(x, t)=0.
(1.1)
In general, these estimates are of interior nature. However, in some cases, they can be extended to be global estimates which hold up to the boundary. First, we will prove the following lemma which is essential in the derivation of our gradient estimates. LEMMA 1.1. Let f(x, t) be a smooth function defined on Mx[0, oo) satisfying ( A  ~ t t ) f = IVfl2+q, (1.2)
where q is a C2 function defined on Mx(0, oo). For any gioen a>l, the function F = t(IVfl2aftaCl)
(1.3)
(3) H. Wu informed the authors that the b~ estimate for compact manifolds without boundary was proved by M. Gromov and S. Gallot in "Structures M~triques pour les Varirtrs Riemanniennes" (1981) and C. R. Acad. Sci. Paris, 296 (1981), 333336 and 365368, respectively.
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P E T E R LI A N D S H I N G T U N G YAU
satisfies the inequality (AO) F>'2(Vf' VF)I
(1.4)
+ 2t ( l V f l 2  f t  q ) 2  a t A q  2 ( a  1) t(Vf, Vg),
n
where K(x), with K(x)~O, is a lower bound of the Ricci curvature tensor of M at the point x E M, and the subscript t denotes partial differentiation with respect to the tvariable. Proof. Let e~, e2 ..... e, be a local orthonormal frame field on M. We adopt the notation that subscripts in i, j, and k, with l<~i,j, k<.n, mean covariant differentiations in the e;, ej and e, directions respectively. Differentiating (1.3) in the direction of e;, we have Fi = t(2fj fjiaftiaqi),
where the summation convention is adopted on repeated indices. Differentiating once more in the e; direction and summing over i= l, 2 ..... n, we obtain
AF = t(2fj~+ 2fjfjiiaftiiaqii)
>~t [ 2 (Af)2+2( Vf, VAf)2K[VflZa(Af)taAq], where we have used the inequalities
i,j
and
n
fJ~. = f~fi~j+ReL fj~ (Vf, VAf)KlVfl 2.
Applying the formula
A f = IVfl2 +q +f, =  1 F  ( a t
1) (q +ft),
we conclude,
A F ~ 2t(Ivf[2ftq)Z2(Vf, V F )  2 ( a  1 ) t ( V f , V(ft) )
n
 2 ( a  1) t(Vf, Vq) 2KtlVfl2+aFta(lVf[Zaftaq)
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+ a ( a  l) tftt+a(a 1) t q t  a t A q = 2 t (iVfl2_ft_ q) 2_2 ( Vf, VF) +F,(IVfl 2 a f t  a q )
n

2Kt]Vfl2, atA q  2 ( a  l) t ( Vf, Vq).
This proves the lemma. THEOREM 1.1. Let M be a compact manifold with nonnegative Ricci curvature. Suppose the boundary o f M is convex, i.e. II~>0, whenever OM~:~). Let u(x, t) be a nonnegative solution o f the heat equation
.
/,  o
on Mx(O, ~), with Neumann boundary condition
=0
Ou Ov
on OMx(O, ~). Then u satisfies the estimate u2 on Mx(0, oo). Proof. By settingf=log (u+e) for e>0, one verifies that f satisfies u 2t
Applying Lemma 1.1 t o f b y setting a = l , q=0, and K=0, we have
,1,,
The theorem claims that F is at most n/2. If not, at the maximum point (Xo, to) of F on M x [0, T] for some T>0,
F(x~ to) > 2 > O.
Clearly, to>0, because F(x, 0)=0. If Xo is an interior point of M, then by the fact that (Xo, to) is a maximum point of F in Mx[0, T], we have
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PETER LI AND SHING TUNG YAU
AF(xo, to) ~<O,
VF(xo, to) = O,
and
Ft(xo, to) >~O.
Combining with (1.5), this implies 0 ~ 2 F(xo, to)(F(xo, t o )  2 ) , >
nt o
which is a contradiction. Hence Xo must be on aM. In this case, since F satisfies (1.5), the strong maximum principle yields
~vOF(x0, toO> 0.
However
nI
(1.6)
aF = 2 f i f _f~ = 2 E L
OV a=l
fa,,
since fv=uJ(u+e)=O on aM, and we are assuming that e.=a/av. Computing fay in terms of the second fundamental form II=(ho,a), we conclude that aF av nI
2 E
a,/]=l
h~ fa f~ =  2II(Vf, Vf).
Inequality (1.6) and the convexity assumption on aM yield a contradiction. Hence F~<~, and the theorem follows by letting e~0. THEOREM 1.2. Let M be a complete manifold with boundary, aM. Assume p s and let Bp(2R) to be a geodesic ball o f radius 2R around p which does not intersect aM. We denote K(2R), with K(2R)>~O, to be a lower bound o f the Ricci curvature on Bp(2R). Let q(x, t) be a function defined on M? [0, T] which is C 2 in the xvariable and C ~ in the tvariable. Assume that
n
Aq <~0(2R) and
IVql ~<r,(2R)
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Bp(2R)? [0, T] for some constants O(2R) and y(2R). If u(x, t) is a positive solution of the equation
on
Aq
u(x,t)=O
on Mx(O, T], then for any a > l and eE(O, I), u(x, t) satisfies the estimate
IVU]2u T 2
C3a2R_2(l+ev~+cx2(a_l)_l)+2a2t_l
_2 _ + [ C4(~4(ct_ 1)2a4et)ta+2(le)la'(a1)2K2+2a'O ] !/2 j
on Bp(R), where C3 and C4 are constants depending only on n. Proof. As in the proof of Theorem 1. l, we define the function F(x, t) = t([Vfl2aftaq)
where
f = log u.
Let ~(r) be a C 2 function defined on [0, oo) such that q~(r) = { 10 if rE[0, 1] if rE [2, oo),
with
~31/2(r)
c I,
and
~"(r) ~  C 2 ,
for some constants C~, C2>0. If r(x) denotes the distance beween p and x, we set
We consider the function ~0F, with support in Bp(2R)x [0, 0o), which is in general, only Lipschitz since r(x) is only Lipschitz on the cut locus of p. However, an argument of Calabi, which was also used in [8], allows us to assume without loss of generality that ffF is smooth when applying the maximum principle.
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P E T E R LI A N D S H I N G T U N G YAU
Let (Xo, to) be a point in M x [0, T] at which 9 F achieves its maximum. Clearly, we may assume 9 F is positive at (Xo, to), or else the theorem follows trivially. At (Xo, to), we have
V(gF) = O,
a(gF) t> O,
at and
A(grO < 0.
By a comparison theorem in Riemannian geometry, Aq~ = q3'Ar + ~"[Vr[2 R R2 >>' Cl (n1) M/K Applying Lemma 1 to the equation A(tpF) = (Ag) F + 2 (V 9, VF) + 9(AF), and using the above inequality, we arrive at A(gF) I>  F(C I R  l(n  1) ~ coth (R X/K")+ C 2R 2) C2 R 2"
+2(V9, V(gF)) 9  1 + 9
r/
lFt2 (Vf,VF)
+ 2 t([Vf] 2_ft_ q)2_ t  t F  2Kt[Vf[ 2 atAq2(aEvaluating at (Xo, to) yields 0 >I  F ( C 1Rl(n  1) X/K coth (R X/K)+ C2 R 2) f +2F(Vf, Vq~)+ 2 to cp(fV I 2  f t  q ) 2  g t  I F 2Kto 1) t(Vf, Vq) ] 2/~V~129 1.
lVfl:a t0 A q  E ( a 
l) t o9 ( Vf, Vq).
Multiplying through by 9to and using the assumptions on Aq and IVql with the estimate on IV~I, this becomes
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0 ~ toq)F['C, R'(n l) V~" coth (R V ~ ' )  C 2R22C~ R2to l]

2to cpF1VflC, R'q~'/2 + 2 ~ ~o2[(ivfl2 _f,_ q)2_ nKIVfl2]
n
(1.7)
 a ~ O2(a 1) to rC/21vf I. 2
ff we let
y= r
~
and z = q~(ft+q), and observe that C 1R 1 ( n  l) V K coth (R V'K) + C 2R2+2C~ R 2 ~<C 3R2(1 +R V ~ ) , for some constant C3 depending only on n, (1.7) takes the form
0 >>q)F[t oC3R2(1 +R V ~ )  l] + 2 t 2 ? [(yz)2nC! Rlyl/2(yaz)nKyn(a 1) yyU2]_ ~ aO.
On the other hand, we observe that (1.8)
(yz)EnCl RlyU2(yaz)nKyn(a 1) yyU2
= (1  e  b ) y2(2ea) yz+z2+(eynC1 Rly u2) (yaz)+6y2+nKyn(a  l) yyV2 (1.9)
=(a'2)(yctz)2+(1et~a'+2)Y2+(1a+2a2)zZ +(ey nC1RlyU2) (y_az) +6yznKyn(a 1)),yl/2.
Setting 6 = ( a  1  1 ) 2 and e=22al2(a~l) 2, we check that l_e_6_al+_._e = 0 2 and
1   a + ~   C t 2 = 0.
Hence, (1.9) becomes
(yz)2nCl RlyV2(yaz)nKyn(a  1) yyV2 >ta2(yaz)2C3 a2(a_ 1)lR2(yaz)+ a2(a 1 ) 2 y 2  n K y  n ( a  1)yyV2,
(1.1o)
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PETER LI AND SHING TUNG YAU
where we have used the fact that 2ae(a 1) y nC 1Rlylr2 >~ _ ~ C~ a2(a 1) IR ~2
>I  C 3 a 2 ( a  I )  I R 2.
We will estimate the last three terms of (1.10) as follows, a2(a  1)2y2nKyn(a  1) ~V 1/2 ae(a  1)2y2(1  e ) a2 ( a  1)2y2~ (1  e )  I a e ( a  1)2K 2  n ( a  1) 7,y!/2 ~>ea2(a  1)2y2 n(a 1) 7'y!/2~ (1 e) la2(a  l)2K 2
n2   C4(~4(~  l ) 2 a 2 E  !) 1/3~ ~  ( l   e ) n2
n2
I a2(a 
I)2K 2
for any e E (0, 1). Combining this with (1.8) and (1.10), we conclude that 0>t q~F[to C3R2(l + R V ~ ) n
l]
+ 2 [a_2(q~F)2_ C3 a2(a  1)_lR_2q~Fto]
2 ( / _ 2 ( ~ 0 F ) 2 [ C 3 foR_2( 1+ R V ~ ) + a 2 ( a _ n
i)_i)+ 1] (q~F)
This implies that at the maximum point (Xo, to)EMx[0, T], f F ~ C 3 a2to R 2(1 + R ~ / K + a2(a  1) i) + 2 a2
I" 4
(al)
2 4
a :)
!1/3
n
2
I 4
a
2 2
K
_ u
3
"] 1/2
0J
.
In particular, on M x {T}, F satisfies the estimate as claimed in the theorem for a > l and 0 < e < l , since to<~T. THEOREM 1.3. Let M be a complete manifold without boundary. Suppose u(x, t) is a positive solution on Mx(0, T] o f the equation
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(
aq~
a)
u(x, t) = O.
Assume the Ricci curvature o f M is bounded from below by  K , for some constant K>~O. We also assume that there exists a point p E M, a constant O, and a function F(r, t) such that
IVql (x, t) ~<~,(r(x), t)
and Aq<.O on Mx(O, T], where r(x) denotes the distance from p to x. Then the following estimates hold:
(i) I f K=O and lim y(r, t) ~< r(t),
r~ oo r
then u2 on M x (0, T]. ~tl+q+Cs~:as(t)+ u 2 0
(ii) I f y(r, t)<~,o(t) for some function yo(t), then
iVul au,< 2_ u ctq+nct2t_l+C6[Y oa(t)+(a_l)_lK+01/2] u2 2
on M x ( 0 , T]for all a E ( l , 2 ) . Proof. To prove (i), we simply set a  l = R  2 r  l / 2 ( t ) in Theorem 1.2, and let
R~. As of (ii), we just let R>~ without any substitution. THEOREM 1.4. Let M be a compact manifold with Ricci curvature bounded from below by  K , for some constant K>~O. We assume that the boundary o f M is convex, i.e. II~>0. I f u(x, t) is a positive solution on M x ( 0 , oo) o f the heat equation
with Neumann boundary condition au Ov = 0
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PETER LI AND SHING TUNG YAU
on a M x [0, o0), then u(x, t) satisfies
IVul
U2


au, ~< n
U
Ct2(Ct 1)lK+ 2
Ct2tI
on Mx(0, ~), for all a > l . Proof. This follows by combining the arguments in the proofs of Theorem 1.1 and Theorem 1.3.
COROLLARY 1.1. Let M be a complete manifold without boundary. Suppose the Ricci curvature o f M is nonnegative, and suppose q is a C2 function defined on M with Aq~<0
and
IVql =
o(r(x)),
where r(x) is the distance from x to some fixed point p E M. Ifinf q<O, then the equation
( A  q ) u(x) = 0
does not admit a positive solution on M. In particular,
infq = inf {Spec (Aq)},
where Spec ( A  q ) denotes the spectrum of the operator A  q . Proof. Let u(x) be a positive solution of ( A  q ) u=0. Applying Theorem 1.3 (i) to
this time independent solution, we arrive at the estimate
IVul u2
n ~+q"
Letting t~, and evaluating at a point where q<0, we have a contradiction, unless infq~>0. To prove the second half of the corollary, one observes that the quadratic form associated to A  q is given by
f u.fqu
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which is clearly bounded from below by infq. Hence inf {Spec (Aq)} I> infq. On the other hand, we know that for e>0, the equation
[A  ( q  i n f qe)] u = 0
has no positive solution, which implies inf {Spec (Aq)} ~<infq+e (see [10]). However, e is arbitrary, which yields the desired equality. COROLLARY 1.2. Let M be a complete manifold without boundary. Suppose the
Ricci curvature o f M is bounded from below by  K , for some constant K>O. Assume that q is a C 2 function on M with Aq<~O and
IVql (x) ~ r(rQg, x)),
for some constant O, and some function y depending only on the distance, r(p, x), to some fixed point p E M. Then
infq ~<inf {Spec (Aq)}
and
inf {Spec ( a  q ) } < Q,
where Q is finite and is defined in the following cases:
(i) I f K=O, and limr_,~ rly(r)<~v for some constant r, then
Q=infq+Cs~
"3
+~yO)
[ n
\1/2
.
(ii) I f ~(r)~yofor some constant Yo, then
Q = a infq+C6[y~o3+(a  1)IK+O 1/2] for a E(1,2).
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PETER LI AND SHING TUNG YAU
Proof. Following the proof of Corollary 1.1, we apply Theorem 1.3 to any positive solution q0 of the equation
( A  q ) q0= gcp for 2>Q.
w2. Harnack inequalities We will utilize the gradient estimate in w 1 to obtain Harnack inequalities for positive solutions of (1.1). THEOREM 2.1. Let M be a complete manifold with boundary, aM. Assume p E M and let Bp(2R) be a geodesic ball o f radius 2R centered at p which does not intersect aM. We denote K(2R), with K(2R)~>0, to be a lower bound o f the Ricci curvature on Bp(2R). Let q(x, t) be a function defined on M x [0, T] which is C 2 in the xvariable and C l in the tvariable. Assume that Aq ~<0(2R)
and
[Vql ~<y(2R)
on Bp(2R)? [0, T] for some constants 0(2R) and y(2R). I f u(x, t) is a positive solution o f the equation Aq~ u(x,t)=O
on Mx(0, T], then for any a > l , O<h<t2<T, and x, y EBp(R), we have the inequality u(x, t,) <~u(y, t 2) \ tl / where
exp (A(t2tO+Oam(x, y, t2tO),
A = C7[aR 1V~+a3(a  1)lR2+y2/3(a  1)l/3al/3+(aO)l/2+a(a  1)lK]
and
~a,R(x,y, t2tt) = inf ~ ye r~R) 4(t 2 t l)
(
fo
1~12ds+(t2tl)
fo
q(y(s), ( l  s ) t2+st Ods
)
with inf taken over all paths in Bp(R) parametrized by [0, 1] joining y to x.
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Proof. Let 7 be any curve given by 7: [0, 1]>Bp(R), with 7(0)=y and 7(1)=x. We define r/: [0, 1]~Bp(R)x[h, t2] by
rl(s) = (~'(s), (1 s) t2+Stl). Clearly r/(0)=(y, t2) and r/(1)=(x, t0. Integrating (d/ds)(logu) along r/, we get
logu(x, tt)logu(y, t2)=fol(~slogu)ds
= { (~,, V(log u)) (t2t I) (log u)t} ds.
Applying Theorem 1.2 to (log u)t, this yields
l~
u(y, tz) /
fo'
(2.1)
Viewing IVlog u[ as a variable and the integrand as a quadratic in IVlog u[, we observe that it can be dominated from above by
al~[2 +(t2t,) IA+2at~ +q]. 4(tEtl)
Since t=(1s) t2+Stl, (2. l) gives
(u(x,t,)~ ' na t2 log \ u  ~ ,t2)/~< f0 [.~4(t2_alTltl +(t2 t 1)q(y(s),(1s)t2+stl)}ds+2log(~l)+a(t2tl). )
The theorem follows by taking exponentials of the above inequality. Obviously, applying Theorems 1.3 and 1.4 instead, the above method yields: THEOREM 2.2. Let M be a complete manifold without boundary. Suppose u(x, t) is a positive solution on Mx(0, T] of the equation ( A  q  ~ t t ) u(x, t) = O.
Assume the Ricci curvature of M is bounded from below by K, for some constant K>~O. We also assume that there exists a point p E M, a constant O, and a function ),(r, t), such that
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PETER LI AND SHING TUNG YAU
IVql (x, t) <~y(r(x), t)
and Aq<.O on M x (0, T], where r(x) denotes the distance from p to x. Then for any points x, y E M, and O<tl<tE<~T, th e following estimates are valid:
(i) / f K = 0 and lim y(r, t) ~< r r~ r oo
for all t E [0, T], then
/ t \ n/2
u(x, t l ) ~ u(y, t 2 ) / ~ l )
where
exp(Cs(r2/3+ 01/2) (t2tl)+Q(x, y, t2tl)),
O,xyt t, inf{' foli'12ds+(t2tl)foq(~'(s)'(1s)t2+Stl)dS'!
rer 4(t2t1)
with inf taken over all paths in M parametrized by [0, 1] joining y to x.
(ii) I f y(r, t)<~yofor some constant Yo in M x [ 0 , T], then
/ t \ nod2
U(X, tl)<~u(y, t 2 ) / ~ l )
for all a E (1,2), where Q~(x, y, t 2  t I) = inf
exp(C6(t2tl)(y2/3+ol/2+(~l)lg)+Qa( x, Y, t2tl))
yer
{
9 ~
4(t2t 1)
fo
1~,[2ds+(t2tl)
fo
q(y(s), ( l  s ) t2+st 1) ds .
}
THEOREM 2.3. Let M be a compact manifold with Ricci curvature bounded from
below by  K , for some constant K~O. We assume that the boundary of M is convex, i.e. 111>0. Let u(x, t) be a positive solution on Mx(0, oo) o f the heat equation
with Neumann boundary condition
au m= Ov
0
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on OMx[0, oo). Then for any a > l , x, y E M , and 0<tl<t2, we have
/ t \ hal2
u(x,t,)<~u(y, t2)t~ )
exp(~22 ( a  l '  ' K ( t 2  t , ' d ar2(x'Y)]
where r(x, y) is the distance between x and y.
A mean value type inequality can be easily derived by averaging the function over any set in either the xvariable or the yvariable. In fact, this will be the form which we utilize most in the latter sections. For example, a corresponding mean value inequality of Theorem 2.3 will read <~ /~) exp(~22 (al)tK(t2"t')4
4(t2t,)]
We also remark that from the definitions of O's, they clearly satisfy the following relations:
Pa, 
y, t2ti) = Oa(x, Y, t2tl),
(2.2) (2.3)
QI, ~(x, y, t2t 1) = Ol(X, y, t2t 0 = e(x, Y, t2tl),
and
Qa(x, Y, t2t~) = ar2(x, y) 4(tz_t3)
if q0.
(2.4)
COROLLARY 2.1. Let M be a compact manifold with nonnegative Ricci curva
ture. I f 8 M ~ , must satisfy
we assume that it must be convex. The Neumann heat kernel on M
H(x, y, t) >i (4zrt) n/2 exp ( r2(x' y) ~. \ / 4t In particular, the Neumann eigenvalues /ti of M satisfy
~
i=0
e~, ~' I> (4~t)nl2V(M).
Proof. Apply Theorem 2.3 to the function u(y, t) = H(x, y, t)
12868283 Acta Mathematica 156. lmprim6 le 15 mai 1986
170 gives
PETER LI AND SHING TUNG YAU
H(x,x, tO<.H(x,y, t2)
Since
exp \ ~ ] .
H(x, x, tl)~
( 4 ~ t l ) n/2
as t~)O estimate of H(x, y, t) follows. To prove the estimate of the theta function, the we simply integrate H(x, x, t) over M.
w3. Upper bounds of fundamental solutions In this section, we will derive upper estimates of any positive of the equation
L2
fundamental solution
Aq~ u(x,O=O,
where we will assume the potential function q is C 2 and is a function on M alone. We recall again the definitions of O's, since q is timeindependent, they can be written as
Oa,R(x, y, t) = inf ~ a r ~r(R) [ 4t
l#12ds+t
q(y(s)) ds
(3.1)
where F(R) = {~,: [0, 1] ~ B p ( R ) I~,(0)=y, y(1) =x}. Moreover
qa, 
where F(oo)=M, and
y, t) = qa(x, y, t),
(3.2)
01. 
y, t) = el(x, Y, t) = q(x, Y, t).
(3.3)
We remark that when q~>0, Q is a metric on M, though 0 might not be a distance function. Abusing this term, we will refer to 0 as "the metric" even when q is sometimes not assumed to be nonnegative. The following discussion of 0 is classical, especially among physicists, hence details of proofs will be omitted. If ~, is a minimizing curve for O(x, y, t), considering a compact perturbation of y, one computes that the geodesic equation of 0 is given by V~, = 2t2 Vq. (3.4)
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
171
Taking the inner product with ~,, (3.4) gives
ds (1~'12)= 2(V~,~,, ~,) = 4tZ(Vq, ~,).
Integrating along y from 0 to s, we have li,12(s)1#12(0)= 4t2(q(?(s))qO,(O))). Hence
d
[$'12(s)4tZq(y(s)) = {1;'12(O)4t2q(y(O)),
(3.5)
for all s E [0, 1]. Ifx is not a "cutpoint" of y, we can find a 1parameter family of curves y,, joining o(r) to y, where o: (e, e),M such that o(O)=x. We compute
1
d =
I
d?
drd OIt=~ 2t = l < # ( 1 ) '  ~  r (1'>
~=o~'tJo \ ['/V~'~" d}'~' +t Jo ['<Vq,~r> ,:=o dr/I,=o
=yi\r
1 /.(1), dT(1) \
/ r=o'
VxO(x, y, t) = l ~ ( 1 )
after using (3.4). We conclude that (3.6)
and
IVxOI2(x,y, t) = ~
Similarly, we compute
]~'lz(1).
Ot (x, y, t) = ~
1 4t2
4t2
li,12+t
q(y)
foi l~'12+1 '
I~'12+
l
d 1
_if,[, q(~,),
where we have used (3.4), and the assumption that ?t(O)=y, and FAD=x, for all t.
172
PETER L I AND S H I N G T U N G YAU
However, by (3.5), we derive
aO (x,y,t)=
at
I ~
1~,12(1)+q(x).
Together with (3.6), we have proved the following: LEMMA 3.1. IVxel2(x, y, t)+ ~ (x, y, t) = q(x) ~t
and
IV~ol2(x, y, t)+~t (x, y, t) = q(y).
We remark that it is also well known that the function g is Lipschitz on M (see Appendix). In particular, the above lemma is valid in the weak sense on M. Let us define the function
g(x, y, t) = 20(x, y, (1+26) Tt)
for x, y E M and 0~<t~<(1+26) T, where 6>0. Lemma 3.1 implies that g satisfies 89 gl2+gtEq(y) = 0 IVy weakly.
(3.7)
(3.8)
When q  0 , we may assume aM=t=~ but convex. Since in this case g(x, y, t) is just a multiple of the square of the distance function r(x, y), (3.8) is still valid due to the assumption on aM being convex. LEMMA 3.2. Let M be a complete manifold which can be either compact or noncompact. Suppose H(x, y, t) is the fundamental solution of(1. i) on M x M x [0, oo)./f
qO, we may assume aM~(~ but convex, and H(x, y, t) satisfies either the Dirichlet or the Neumann boundary condition on aM. Let
F(y, t)
f H(y, z, t) H(x, z, T) dz
dS I
(3.9)
for x E M , SI~_M, and 0<t~<r<(l+26) T. Then for any subset S2~_M, we have fs F2(z, r)dz<' fs H2(x,z,T)dxsupexp(2Q(x,z, (l +26) T))
2 l zESI
x s u p exp (20(x, z, (1+26) Tr)).
zE S2
ON THE PARABOLIC K E R N E L OF THE SCHRODINGER OPERATOR
173
In particular,whenq=O,and S2=B,,(R),we have
f~
x(R) J St
/ :(x, s,) ~
R2
Proof. Since F satisfies (1.1), we consider 0=2 r Bx(k)
Ay
q(y) F(y,t),
(3.10)
where q0(y)=q~(r(x, y)) is a cutoff function of the distance to x alone such that q~(Y) = 10 on outside Bx(2k),
and IVq0[~<3/k.Integrating the right hand side of (3.10) by parts and using the boundary condition on H, we get
O=4fffq~egF(VcP, VF)2fo~fcp2egF(Vg,VF)
2fo'f +fo'f,,2e'F2g, e',Ve,
fMqJ2e~F2,=+fM~2e~F ,=o2fotfcp2egF2q9 2
By the Schwarz inequality,
(3.11)
2fo'fe'F<Vg,
Combining this with (3.8) and (3.10), we deduce that
O<~4fffcPegF(Vq~,VF)fMqflegF2 +~ ~ 2 e g F 2
/=l"
.IM
.
t=O
Letting k+oo, since [Vq01~<3/k, the first term on the right hand side of the above inequality vanishes by virtue of the fact that its integrand is L 2. Hence,
f exp(g(x,Y,r))F2(y,r)dy<~fMexp(g(x,y,O))F2(y,O)dy.
Observing that
174
P E T E R LI A N D S H I N G T U N G YAU
fH(x, y,
F(y, 0) = ~0 and (3.7),
T)
if y E S l if y r S~,
f exp(g(x,y,O))F2(y,O)dy<~supexp(20(x,z,(l+26)T))fsH2(x,z,T)dz.
zESI 1
On the other hand, f u e x p (g(x, y, r)) F2(y, r) dy >>./ exp (g(x, y, r)) F2(y, r) dy
JS 2
~> inf exp (20(x, z, (1 +26) Tr)) [ F2(y, r) dy.
z E 82
JS2
This proves the lemma. It is now convenient for us to introduce the following notations: We define
O~(x, S, t) = sup O~(x' z, t)
zES
and
_Qa(x, S, t) = inf Oa(x, z, t)
zES
for any subset S~_M. THEOREM 3.1. Let M be a complete manifold without boundary. Assume the Ricci curvature of M is bounded from below by  K , for some constant K>>.O. We also assume that there exists a point p E M, a constant 0, and a function y(r), such that [Vq[ (x) ~<?(r(p, x)) and Aq<~O on M. Then for x, y E M and t E (0, oo), the following estimates are valid: (i) l f K=O, and lim ~,(r) = r, r, r
ON THE PARABOLIC KERNEL OF THE SCHR(~DINGER OPERATOR
175
then H(x, y, t) <~(I +6)nvI/2(SI) VIn(S2)exp (C5(g213+0112) 6(I+6) t)
xexp (20(x, S 2, 6(1 +6) t)) exp (0(Y, Si, 60) ? (Q(x, $1, (I+26)(1+6) t))
for any 6>0, and any subsets Sl, S2~_M whose volumes V(SO and V(S2) are.finite. (ii) If F(r)<~yo,for some constant Fo, then H(x, y, t) <<.(1 +6)navI/2(SI) VII2(s2) e x p
[C6(Y02/3+ 01/2 + ( a 
1)tK) 6(2+6) t
+~a(x, S 2, 6(I +6) t)+0~(Y. S t, 6t)+O(x, S 2, 6(1 +6) t) _~(x, S~, (1 +26) (1 +6) t)],
for any a E (1,2), 6>0, and any subsets $1, S2c_M with finite volumes. Proof. We will only prove (i), while the proof of (ii) follows similarly by using Corollary 2.3 instead of Corollary 2.2. To prove (i), we apply Theorem 2.2 to the function F(y, t) of Lemma 3.2. This yields
(fs H2(x, z, T)dz) 2= F2(x, T) ~< +6) n exp [2C5(t~ + 0 I/2)6T+20(x, S 2, 6T) (I
+ 20(x, S 2, 6T)2Q(x, S i(1 +26) T)] ~s~H2(x' z, T) dz V~($2),
by setting r=(1+6)T in Lemma 3.2. Applying Theorem 2.2 to the function H(x, z, T) and setting T=(I+6)t, we obtain
H2(x, y, t) <<. +6) 2nexp [2C5(rv3+ 0 v2) 6(2+6) t (1 +40(x, S 2, 6(1+6) t)+ 20(x, Sm, 6t)
 2~__ $1, (1 +26) (1 +6) t)] Vl(Sl) Vl(S2). (x, The theorem follows by taking square root of both sides. COROLLARY 3.1. Let M be a complete manifoM without boundary. If the Ricci curvature of M is bounded from below by  K , for some constant K>.O, then for l < a < 2
176
P E T E R LI A N D S H I N G T U N G YAU
and 0<e<l, the heat kernel satisfies [H(x, y, t) <~C(e)~Vl/2(Bx("v/f))Vl~(By(Vi)) exp/C7 e(a 1)lKt
L
r~(x, Y)
]
The constant C 7 depends only on n, while C(e) depends on e with C(e)>oo as e>O. When K=0, the above estimate, after letting a>l, can be written as H(x, y, t) < C(e) Vl(Bx(Vri)) exp [r2(x'(4+e) ]" y)t Proof. Setting y0=0=0, S 1=By(X/T), and S2=Bx(~/'i) in the estimate of Theorem 3.1 (ii), we have H(x, y, t) <~(I +6)"aVI/2(Bx(V~)) VI/2(Br(V~))
xexp [C6(a 1)1K6(2+6) t+2Oa(x, Bx(V~), 6(2+6) t) +0a(Y, By(~/7), 6t)_Q(x, Br(~/), (1 +26) (1 +6) t)]. Since q~0. 2Oa(x, B~(V'7), 6(2+6) t) = Similarly
a
sup zeBx(x/7 26(2+6) t )
ar~(x, z) _
a
26(2+6)
(3.12)
do(y, B,(VY), 60  46
(3.13)
and
O(x, Br(V'i), (1 +26) (1 + 6) t) = inf rZ(x, z) zeSt(x/7) 4(1 +26) (1 +6) t"
(3.14)
If x E By(V'T), then
O(x, By(Vf),(l+26)(l+6)t)=O>.. r2(x,y)
4t
1
4
(3. ! 5)
On the other hand, if x r
i.e. r(x, y)>X//, we have
p(x, By(V~'), (1 + 26) (1 + ~) t) = (r(x, y)" VT) 2 4(1+26) (1+6) t"
(3.16)
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
177
Applying the inequality (r(x, y )  V " T ) 2 ~> rz(x' y) 1+8 and setting 4(1+28) (1 +8)2=4+e, (3.16) becomes e(x, By(V't), ( 1+ 28) (1 + 8) t) I> rz(x' y)
(4+e) t
t 8'
I+6
4e6 "
In any case, this together with (3.12), (3.13), and (3.15), proves the first estimate as claimed, To show the second estimate, we apply a volume comparison theorem (see [5]), which states that if O<RI<R2<R3, then
V(Bx(R2)) V(K, R 2) V(Bx(RI)) ~< V(K, RI)
and
(3.17)
V(Bx(R3)Bx(R2)) <~ V(K, R3)  V(K, R z)
V(Bx(R1)) V(K, Rl)
'
where V(K, R) is the volume of the geodesic ball of radius R in the constant K/(n I) sectional curvature space form. To estimate V(Bx(V:i)) by V(By(~:i)), we consider the following cases:
(a) If VT>2r(x, y), then
V(Bx(V'i)) V(nx(VSr(x, y)))
V(O, ~/"ir(x, y))
V(O, vi)
<~V(By(V'i)) : ~ ]n \ ~  r(x, y) /
<~ 2"/zV(By(Vf)).
(b) If V'/<~2r(x, y), then
V(BI(Vt)) <~V(Bx(V,i/4)) V(O,vi) v(o, VY/4)
<~ V(By(r(x,y) +V:"//4))(4n) <~ 4nV(By(Vi/4)) r(x,y)+V7/4) V(0, V(O,k/]/4)
<<4nV(By(V,i)) : 4r(x, y)+ V'f.~". \ vi/
178 Hence
PETER LI AND SHING TUNG YAU
:4 r(x, y) + / r2(x, y) '~ H(x'y't)<~ C(e)VI(Bx(V'i))\ ~ l ) n e x p k (4+e)t ]
exp
/ r2(x, y)
by readjusting the constant C(e). Now setting 4+2e to be 4+e, we also derive our estimate as claimed. It is by now clear that the following theorems follow in exactly the same manner as Theorem 3.1 and Corollary 3. I. Of course, in each case, one uses Theorems 2.3 or 2.1 instead. THEOREM 3.2. Let M be a compact manifold with Ricci curvature bounded from below by  K , for some constant K>~O. We assume that the boundary o f M is convex, i.e. 111>0. Then the fundamental solution H(x,y, t) o f the heat equation
with Neumann boundary condition
On
Ov must satisfy
=0,
r H(x, y, t) <~C(e)aVl/2(Bx(V'i)) V J/2(By('V"~'))exp [C 7e ( a  1)lKt
?(x, Y) ]
for all l < a < 2 and 0 < e < l , where the constant C(e))oo as e)O. When K=O, after letting a>l, this estimate can be written as /  r2(x, y) H(x,y, t) <~C(e) VI(Bx(V"7")) exp k (T~e) t ]"
THEOREM 3.3. Let M be a complete manifold without boundary. Assume p E M and let Bp(2R) be the geodesic ball o f radius 2R centered at p. We denote K(2R), with K(2R)>>O, by a lower bound o f the Ricci curvature on Bp(2R). We also assume that q is a C2 function on M with Aq ~<0(2R)
and
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
179
IVql ~<~,(2R)
on Bp(2R). Then for any ct>l, x, yEBp(R), $1 and $2 any subsets of Bp(R), a fundamental solution H(x, y, t) o f the equation
( A  q  ~ t t ) u(x, t) = O
must satisfy H(x, y, t) <~ (1 +6)"VIrZ(Si) VlrZ(S2) exp (A6(2+6) t)
x exp [0a,~(x, $2,6(1+6) t)+Oa,R(y, Si, 6t) 0(x, S 1, (1+26) (1+6) t)+O(x, S 2, 6(1+6) t)]
where
A = CT[aR 1X/K+a3(a  l)IR2+yZ3(ct  1)v3av3+(aO)V2+a(a  1)lK]. The estimate given by Theorem 3.1 can be written in a more comprehensible form when the potential is nonnegative. In fact in this case, we see that
O(x, y, t) ~ r2(x' y) 4t
This ensures that Q(x, y, t) is a proper function in the yvariable. COROLLARY 3.2. Let M and q satisfy the hypothesis o f Theorem 3.1. We also assume that q is nonnegatioe. The following estimates hold: if(i) o f Theorem 3.1 is valid, then for all a>0,
H(x, y, t) <~ ca(e) Vv2(Sa(x , t)) VIrZ(Sa(Y, t))
Xexp [C s e(r2/s+ 0 v2) t(1 +e)lO(x, y, t)],
for all 0<e<89
/f (ii) o f Theorem 3.1 is valid, then
H(x, y, t)<~ C(e)~ t))
vll2(Sa(Y,

t))
x exp
for all 0<e<89 and 1<~a<~2.
[C6(~o3+01~+(a
1)~K) t(1 +e)~0(x, y, t)],
180
PETER LI AND SHING TUNG YAU
In both cases, C(e) is a constant depending on n and e with C(e)~ as e*O, and Sa(x, t)={zE mlQ(x, z, t)<.a}. Proof. We will only give the proof of (ii), while (i) follows identically. By the nonnegativity of q, we observe that if q<~tz, then O~(x,y, t,) = inf~ qa y [ 4tin
f0 f0)
[~,12+tl q
~lf } 'J 3o q (3.18)
= i n f ~ tit2 (~t2 fo I [~,12+t2fo I q )  f a t ~  t t t, \ t, = i nyf ~t a tt, ( ~ t 2 fol 1~'12+t2 f0' q ) } 2 =
ctt 2 tl
O(x, y, t2),
for all a~>l. By Theorem 3.1 (ii), if we set Sl=Sa(Y, t) and S2~Sa(x, t), we only need to estimate the following:
Oa(x, S a(x, t), 6(1+6) t) ~<
6(I+6)
O(x, Sa(X, t), t) << 
(ta
6(1+6)'
and similarly Oa(Y, Sa(Y, t), 60 ~< aa 6' and
~(X, Sa(x , t), 6(1 +6) t) ~<
6(1+6) "
Finally _O(x,Sa(y, t), ( 1+ 26) ( 1+ 6) t) ~> (1 + 26) (1 + 6) '0(x' Sa(Y' t), t). If x E Sa(y, t), then we observe that
_O(x, Sa(Y, t), t) ~ 0 > O(x, Y, t) 1.
I
On the other hand, if x ~ Sa(Y, t), then for any z E Sa(Y, t), we claim that 0(x, z, t) ~>? y, (1 +e) t)O(y, z, et). (3.19)
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
181
Indeed, if )'! and )'2 are the minimizing curves for O(z, x, t) and 0(Y, z, t) respectively, we reparametrize )'1 U)'2=)' defined by
y(s) = f )'l((1 ~
+e) $),
if 0_.< s <   .  l+e ifl1 ..< s ...<1. l+e
1
L)'2((l+e)else~),
Clearly y(0)=x and y(1)=y, hence
O(x,Y, (l+e) t) ~ +e)~ < 4(1
 4(l+e)t fo' l+e
l~12+(l+e) t
Y0
qO'(s))
(l+e)t I~d2~ l + e fo'q()'O
q()'2)
4 (l+e) e' f0' ]~212.F1 l +e) )et 1 fo' 4(l+e)t ( e ( +
= O(x, z, t)+Q(y, z, et).
Therefore (3.19) is valid. To conclude the proof, we simply apply (3.18) again and deduce that p(x, z, t) I> (1 +e)lO(x, y, t)elQ(y, Z, t) /> (1 +e)lQ(x, y, t)ela, for all z E Sa(Y , t).
w4. Lower bounds of fundamental solutions
The goal of this section is to derive lower estimates on positive fundamental solutions of the equation
where q is a C 2 function on M. When q0, CheegerYau [6] proved a lower estimate of the heat kernel in terms of the kernel of a model. In particular, they showed that if the Ricci curvature of M is bounded from below by  K with K~0, then the heat kernel of M is bounded from below by the heat kernel of the constant curvature simply connected space form with
182
PETER LI AND SHING TUNG YAU
sectional curvature identically  ( n  1 )  l K . In Theorem 4.1 below, we will prove an estimate which is different from that of CheegerYau. When K=0, this estimate which we will derive is sharp in order, especially for large t. However, when K>0, our estimate does not seem sharper than that in [6]. In view of this, we will only prove the theorem for K=0. THEOREM 4.1. Let M be a complete manifold without boundary. Suppose the Ricci curvature of M is nonnegative, Then the fundamental solution of the heat equation satisfies
[  r2(x, y) H(x,y, t) >>C'(e) V'(B~(V~/)) exp k (4~e)t J where C(e) depends on e>0 and n with C(e)+oo as e+O. Symmetrizing the above estimate, we also have
H(x, y, t) >~C'(e)
V'~(Bx(VF)) v'~(/~,(VT)) exp / :(~, y) ]. ~, ~
Proof. By Theorem 2.2, we have
/,
(mH(z,Y,(1d)t)~V(Bx(R))H(x,y,t)(1d)"/2exp
(:)
~
.
(4.1)
We will estimate the left hand side of (4.1). Let the function ~(z) be defined as rp(z)=q~(r(x, z)) which is a function of r(x, z) with {~
~(r(x, z)) =
0~<tp~<l, and Ocp/ar<O. If we let
on Bx(X/]=~R) outside B~(R),
F(y, t) = fM q~(r(x, Z)) H(Z, y, t)
be the solution of the heat equation with tp(r(x, y)) as initial condition, then
fs
H(z, y, t) >IF(y, t). x(R)
TO estimate Fly, t) from below, we apply the method of Cheeger and Yau in [6]. We will simply outline the argument as follows:
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
183
Let/~(r, t) be the solution of the heat equation in R" with initial data
t~(r, 0) = ~o(r),
where r is the distance to the origin. Since tp is a function of the distance alone, one verifies that F must also be a function of r for any fixed time. Hence, the notation/~(r, t) is valid. By the argument in [6], we conclude that F(r(x, y), t) ~<F(y, t), provided we can justify the assumption (4.2)
~r(
OP
Or
r, t)<~O
on R"x[0, oo). However, by rotational symmetry of F', we see that (0, t)  0 (4.3)
for all t. Also
aP
07(r, o) _ ~am ~< o, lira ~ r ( r , t ) = 0.
aP
(4.4)
and (4.5)
for all t, since ~p' has compact support. Moreover OF~Or satisfies the differential equation
[~ +(n1)ri~(nl)r2~l<~r
o'_1~aP = O.
Applying the maximum principle for parabolic equations on [0, oo)x [0, oo), and in view of the boundary conditions (4.3), (4.4), and (4.5), we conclude that
aP<~0.
Or
Therefore, (4.2) is valid. Hence
F(y,t)>~(4~tt)~fRcP(l~l)exp(i47~12)dL
184 with
PETER LI AND SHING TUNG YAU
ffl=r(x, y).
Combining with (4.1) and setting R=X/i, we have exp ( lYZl2 ) d~ 4(1 t~) t
H(x,y,t)~C(~)V(Bx(~Cri))tn/2f~
i~<Vg~t
~ C(6) V(Bx(V~i))eXp ( g(llff__~2) ) 9 t
Writing 4  e   4 ( 1  6 ) , the theorem follows. THEOREM 4.2. Let M be a compact manifold with boundary, OM. Suppose the
Ricci curvature of M is nonnegative, and if OM~=(~, we assume that OM is convex, i.e. II~>0. Then the fundamental solution of the heat equation with Neumann boundary condition satisfies H(x,y,t)~CI(e)Vl(Bx(~/[))exp(rE(x'y) )
( 4  e) t
for some constant C(e) depending on e>O and n such that C(e)>ooas e>O.Moreover, by symmetrizing, H(x, y, t)>1 Cl(e) V1;Z(Bx(V'T))V1/2(By(V~))exp ( tg(x' y) ).
(4 e) t
Proof. We can apply Theorem 2.3 to obtain (4.1). Following the notation as in
Theorem 4. l, we only need to show that
F(y, t) >tl~(r(x, y), t).
Their difference,
G(y, t) = F(y, t)l~(r(x, y), t),
satisfies the inequality
(A~) G(y,t)~<0,
in the sense of distributions. We now claim that G (y, t) I> 0 (4.6)
weakly on aM for a dense subset o f x E M. Clearly, to prove the estimate of H(x, y, t), it suffices to prove it on a dense subset of x E M. The general case will follow by passing to the limit.
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
185
Assume (4.6) holds for a particular x EM. To show that G(y, t)~0, we simply consider the function G(y, t), if G(y, t)~<0 G_(y,t)= O, if G(y, t) ~ 0. G_(y, t) is a Lipschitz function on M and it is nonpositive. Hence
s163
The left hand side of (4.7) can be written as
(4.7)
G_(y, t )~tt(Y, t ) dy dt = ~
=T
~
G2 (y, t ) dy dt
f G~ (Y, O) ay
f,G(y,r)ayl f,,, =? fM T)dy.
The last equality follows from the fact that G(y, 0)=0. On the other hand, the right hand side of (4.7) can be calculated as follows:
s
Hence, we have
G_(Y, t)AG(y, t)dydt = 
(VG_(y, t),VG(y, t)) dyde
+s fMG(y,t)~v(y,t)dydt <s
s
Therefore G(y, T)~>0, and since T is arbitrary, this proves the required inequality. The estimate for such a point x will follow from the rest of the argument in Theorem 4.1. To prove the claim that for a dense subset of x E M, (4.6) holds weakly, we observe that since/? is a decreasing function of r(x, y), it suffices to show that
8r(x, y) >>. 0
8v
13868283 Acta Mathematica 156. Imprim6 le 15 mai 1986
(4.8)
186
PETER LI AND SHING TUNG YAU
weakly for a dense set of x E M. With this flexibility of slight perturbation of x, we may assume that the cutlocus of x intersects OM along a set with (n1)measure zero. We denote this set by 5e=_OM. For any yEOM~, there exists a unique geodesic y(s) joining x to y with ~(O)=x and ~(r(x, y))=y. This geodesic is the distance realizing curve because OM is convex. Clearly
Or(x, y) = OrO'(s),y) Ov av '
for all s E [0, r(x, y)]. On the other hand, by the convexity of OM,
ar(y(s), y) I> o
av for s sufficiently close to r(x, y). Hence inequality (4.8) is established for y ~ 5Dwhich was claimed. We will now prove a lower bound for the fundamental solution of the equation
Similar to the upper bound obtained in w3, we have to assume that M has Ricci curvature bounded from below by 0 and Aq is bounded from above.
THEOREM 4.3. Let M be complete manifold without boundary. Assume that the Ricci curvature of M is nonnegative. Suppose q is a C 2function on M with
Aq<~O and
exp (O(x, y, t)) E Le(M).
Then the fundamental solution, H(x, y, t), of the equation Aq~ must satisfy H(x, y, t) >~(4~tt)'a2exp (  ( n~~ l/2tQ(x, y, t)). ) Proof. We will first study the "geodesics" corresponding to the metric O(x, y, t).
To do this, we assume that x is not a cut point of y, that is, for any point z in a
ufx, t)=O,
ONTHEPARABOLIC KERNELOF THESCHRODINGER OPERATOR
neighborhood of x, there exists a unique distance minimizing curve y which gives
187
p(x, z, t) = l fo' [~'[2 fo' q(y(s)). +t
All the theories which we will derive for the metric O(x,y, t) are parallel to the Riemannian situation (q=0). Hence, we will only outline the proofs, and the reader can consult [3] for a more detailed line by line explanation. We recall that by the first variation formula for geodesics, we have the geodesic equation given by (3.4), i.e. Ve~ = 2t2Vq and
V~ O(x, y , t) = ~~7(1). 1 .
(4.9)
(4.10)
The second variation formula for geodesics is given by
020[v~2O~, ' ~IfoI(VvVTV'T)~foI(VTV'VTV))"~I(fo~U
~(Vq,vvv)+ foj (vv(Vq),v)}
(4.11)
Using (4.9) with T~, and the fact that
(Vv(Vq), V)=Hessq(V, V), we
have
~+ ~176 ~o ~1 {~01 (R,~ r, v)+ (v~ v, r) lifo I tVTVl } +t fO1Hessq (V, V). 2 ~ Moreover, by differentiating (4.9), we see that the Jacobi equation is given by
VvVTT=
But
2t2Vv(Vq).
Vv VT T = VT Vv T+RvT T,
188
PETER LI AND SHING TUNG YAU
whence we can write the Jacobi equation as VrVv T = R r v T+2t2 Vv(Vq). (4.12)
If we fix the point y and compute the second derivative of Q as a function of x, then the variational vector field can be taken to satisfy
V(O) = 0
and
Vv v ( 1 )  O.
Then
av 2
wo: {fo
1(tl, v),
V V2)+tfoHOSs V
which is the index form along 7 joining y to x. One checks that the basic index form lemma is still valid (see L em m a 1.21 in [3]). In fact, if 7 has no conjugate points, i.e. if there are no Jacobi fields vanishing at 7(0) and 7(s) for all s E (0, 1], and if V is a Jacobi field along y, then for any arbitrary vector field W along 7 with W(0)= V(0)=0 and WO)=V(1),
I(V, V)<<.I(W, W).
Up to this point, the function q is completely arbitrary. From now on, we will assume that q satisfies
Aq<~O
on M. Moreover, we also assume that M has nonnegative Ricci curvature. In this case, we consider el ..... en, n parallel orthonormal vector fields along y. We define W,~s)=saei with [Wil(1)=1. By the second variation formula and the index form lemma,
n
Vyp(x,y, t) <~E I(Wi, Wi)
i=l
Wi)"~
= 2t
tV~W,I 2 +t
E Hessq(Wi' Wi)
i=l
lifo'
S2~Ric(T, T)+
f'
3o
naES2~2 +t S2~Aq j JO
It'
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
189
net2
Ot
2t(2a1) 2 a + l ~< n [ a z 20t 2 ] "~2] [ 2 a Z l + n(2a1) " Choosing l<a<oo to minimize the right hand side by setting
we have
)]
,,,I,+1 + , .
~L~ T
n =~+(nO~ . ~  / 1/2
This inequality, as it stands, is only valid when y is not a cut point of x. However, following an argument of [5] and [27], and using the fact that (4.10) implies that the gradient of 0 points into the cut locus, inequality (4.13) holds on M in the sense of distributions. To complete the proof of the theorem, we simply compute
~/ n J
(4.13)
(Aq~)(4ztt)n/2exp(,(n~)'/2tO(x,y,t))
~ (4stt) ~/2exp (  ( n~~ l/2t  O(x" y ' t ) ) [ IVO[2 Ao  q + ~t + / nO \) '/2+O,J )
I>0,
in the sense of distributions. Since lim(4ztt)'a2exp~t2},,o
/
/nO\ 11~ tO(x, y, t)) = fx(Y),
it follows from the fact that exp (  0 ) is in L2(M) and from Duhamel's principle that
H(x, y, t) >~(4ztt)m2exp (  ( n~~l/2tO(x, y, t)). )
We point out that the L 2 assumption of exp (  0 ) is rather mild. In particular, if q is
190
PETER LI AND SHING TUNG YAU
bounded from below by a constant q0, then
exp (o(x' y' t)) <~exp (
r2(x'
tq~
which is clearly L 2 on a manifold with nonnegative Ricci curvature.
w5. Heat equation and Green's kernel We will utilize the Harnack inequality, the upper and lower estimates of the heat kernel to derive some properties of the heat equation and Green's kernel on a complete manifold. Later on, we will also apply our upper bound to obtain estimates on eigenvalues and Betti numbers for compact manifolds. THEOREM 5.1. Let M be a complete manifold with Ricci curvature satisfying Ric(x) >~ Cgr2~,x)
for some constant, C9>0, where r(p,x) denotes the distance from x to some fixed point p E M. Then any solution u(x, t) of the heat equation
on M x [0, ~) which is bounded from below is uniquely determined by its initial data u(x, O)= Uo(X). Proof. We may assume, by adding a constant to u(x, t), that u(x, t)~O. Let us first prove that u(x, t) is uniquely determined when Uo(X)O. In this case, we will show that u(x, t)~O.
First, we observe that by defining
u(x, t)
f0,
[ u(x,t1),
if 0 ~ t ~ 1 < <
if t ~ l
on M x [0, oo), v(x, t) is a weak solution of the heat equation
By regularity, v(x, t) must, in fact, be a smooth solution. Applying the Harnack inequality to v(x, t), we conclude that
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
191
to)[t )
where
exp
A(tot)+ 4(tot)/
A <~CT[2aCi912+a3(a 1)lr2(x, p ) + 4 a ( a  1)I C 9 r2(x, p)].
In order to obtain the above estimate of A, we have used the curvature assumption. Setting a = 2 , and l<~t<~to/2, we have
o(x, t) <~o(p , to) C( to) exp ( C( to) rZ(x, p) )
for all x ~ M . Since 0  0 on Mx[0, 1], the above growth estimate is valid on Mx[0, to/2]. Applying the uniqueness theorem in [!5], we conclude that v=0 on Mx[0,to/2]. However to is arbitrary, this shows that v0 on M x [ 0 , ~ ) , and hence u  0 on Mx[0, ~), as claimed. To prove the general case uo(x)>.O, we observe that by the maximum principle, the solution Uk(X, t) of the heat equation with initial data
u~(x, o) = 9k(r(p, x)) uo(x)
satisfies
uk(x, t) ~ u(x, t),
if q~k(r(p, x)) is a cutoff function with 0~<q0k<~l, and ~~ 1, 0,
if r ~ k
ifr~>2k.
We now claim that Uk(X, t)*U(X, t) uniformly on compact subsets of M. Indeed, since O~uk<~u, and by the monotonicity of uk in k, the sequence must converge uniformly on any compact subset to some solution v(x, t) of the heat equation with v(x,O)=uo(x). However, Uk<.U, V<~Uon M? ~). Applying our uniqueness argument to u  v which is a nonnegative solution with initial data (uv)(x, 0)=0, we conclude Uk,U uniformly on compact sets. However, the heat equation is known to preserve L2(M) and is unique in L2(M), whence each of the Uk is uniquely determined. Passing to the limit, so is u. On a complete manifold, one defines the Green's function
G(x, y) =
H(x, y, t) dt
192
PETER LI AND SHING TUNG YAU
if the integral on the right hand side converges. One checks readily, that G is positive and AG(x, y)=cSx(y). THEOREM 5.2. Let M be a complete manifold with nonnegative Ricci curvature. If G(x, y) exists, then there exist constants Clo and Cll depending only on n, such that C1o VI(B~(V"f)) dt<~ G(x, y) ~ C n f : VI(Bx(~J))dt and Clo f  VV2(Bx(X/i)) VV2(By(X/[)) dt <~ G(x, y)
Cll
vl/2(Bx(Vri)) vl/2(By(~f't)) dt
where r=r(x, y). Proof It suffices to show that H(x, y, t) dt <<. Cl2 and
f:
V ~(Bx(~/T)) dt
(5.1)
H(x, y, t)dt <<. Cj2
Indeed, by Theorem 3. I, we have
Vt~(Bx(X/i)) VI/2(By(Vr[)) dt.
(5.2)
G(x,y)= ~ and similarly G(x, y) <~
f: f:
H(x,y,t)dt<~
H(x, y, t) dt+Clo
f: ::~
H(x,y,t)dt+
f;
H(x,y,t)dt
I,' ~(B~(VT)) dt
:o
H(x, y, t) dt+C n
Vl/2(Bx(~/'i))Vl/2(By(Vrf)) dt.
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
193
Moreover, the lower bound of G follows by applying Theorem 4.1 to the inequality
G(x, y) >t
f; H(x, y, t) dt.
To prove (5.1), we apply Theorem 3.1 to get
for2H(x, y, t) dt < Cto for2 V'(Bx(X/~))exp (~t ) dt.
Letting s=r4/t, where r2<~s<~, we have
ff V'(Sx(Vi))exp( )dt= t
On the other hand, the comparison theorem (3.17) yields
~ ds.
(5.3)
ov
Hence (5.3) becomes
,
v(o )

v(o, r
V(B
X
"
for2H(x,y,t)dt<~Clo fr~ vl(Bx(~/s))(~)2nexp(~r2)ds. s However, the function
is bound from above, and the claim follows. The proof of (5.2) is exactly the same. Applying our upper bound of the heat kernel for compact manifolds, we obtain the following eigenvalue estimates.
T H E O R E M 5.3. Let M be a compact manifold with or without boundary. If SM:#(~, we assume that it is convex, i.e. II~>0. Suppose that the Ricci curvature of M is nonnegative. Let {0=po<pl~<p2~<...} be the set of eigenvalues of the Laplacian on M. When OM*(~, we denote the set of Neumann eigenvalues also by {0=fl0<]Al~fl2~,,.} and the set of Dirichlet eigenvalues by {(0<)~.1<~.2~...}. Then there exists a constant C13 depending only on n, such that Cl3(k+ 1)2/" ~k ~ d2
194
PETER LI AND SHING TUNG YAU
and
CI3 k 2/n ~k ~>
a~
for all k>~1, where d is the diameter of M. Proof. Let H(x, y, t) be the appropriate heat kernel. Since the Dirichlet heat kernel is dominated from above by the Neumann heat kernel, for either boundary condition, by Theorem 3.2, we have the estimate H(x, y, t) <~Ci3 Vl(Bx(V'i)).
Integrating both sides and applying (3.17), we have
~ e~': <'Cl3 f~Vl(Bx(V~i)) dt <~cl3 fMf(t)
where
(5.4)
v(o,~_
V_~(B~(d)),
f(t) = J V(O, V t )
if t ~ d if t>~d.
I v'(B~(a)),
Since V(Bx(d))= V(M), (5.4) yields (k+ I) e ~k' <~C13g(O, where
(5.5)
ll,
if t~d.
We multiply both sides by dkt and minimize the function d:g(t) as follows: Due to the fact that d ( e udg(t)) =#k e~dg(t)+e~tg'(t), the function minimizes at to must satisfy
#k g( to) =  g' ( to).
ON THE PARABOLIC KERNEL OF THE $CHRODINGER OPERATOR
195
But g'(t)=O w h e n t>~d, to must be less than or equal to d. Hence
/
d\
n
n/
d\n
and
to= ~u~"
Substituting this value into (5.5) yields k+ 1 <~Ci3(dvZ~k) ~. A similar method gives estimates on 21, also. Remark. Obviously, using the same method as above, one can obtain eigenvalue estimates on compact manifolds with Ricci curvature bounded from below by  K , for some constant K~>0. In fact, the resulting lower bounds for the eigenvalues Ak and/~k depend only on n, K, d and k alone. THEOREM 5.4. Let M be a compact manifold with or without boundary, l f OM4=~, we assume that it is convex. Suppose the Ricci curvature o f M is bounded from below by  K , for some constant K>~O, and also the sectional curvatures o f M are bounded from above by x>0. Let bk be the kth Betti number for either the cohomology group Ilk(M) or the relative cohomology group II~(M, OM), then there exist constants Cl4 and Cl5 depending only on n such that bl ~<Cl4 exp (ClsKd 2) and bk ~<C14 exp (C15(K+~r d 2) for k> l, where d is the diameter o f M. Proof. To prove the estimate on bl, we consider the harmonic representative of elements in H1(M) and H~(M, OM). The first is represented by harmonic 1forms with absolute boundary condition, while the latter is represented by harmonic 1forms with relative boundary condition. However, it was proved in [9] that the heat kernel H~(x, y, t) for 1forms, with either boundary condition, can be dominated by IHl(x, x, t)[ <~n ektH(x, x, t).
n
196
PETER LI AND SHING TUNG YAU
Integrating both sides and applying our estimate for H(x, x, t), yields
bl <~C14 fM Vl(Bx(X/[)) exp (Cl5 Kt).
Setting t = d 2, the desired estimate follows. To prove the estimates for k>~2, we follow the same procedure as above for the heat kernel for kforms. In [9], it was proved that f~l IHk(x, x, t) I can be estimated by
fM IHk(x'x't)l<~(k)e'tlJM H(x,x,t),
where B is a lower bound of the curvature term which arises in the Bochner formula. However, it is known that [12] B can be estimated by K and ~. Hence the estimates are established.
w6. The Sehr6dinger operator
In this section, we will study the fundamental solution Hx(x, y, t) of the opertor
A~2q 8
8t
where q(x) is a fixed potential on M and 2 > 0 is a parameter which is varying. The behavior of H2(x, y, 0 as ~>~ will be studied. In the case when M=R n, it was proved in [22] in relation to semiclassical approximation of multiple wells. Theorem 6.1 gives an asymptotic behavior of H~(x, y, t) on arbitrary complete manifolds which enables one to push the argument in [22] through to the setting of a general manifold. THEOREM 6.1. Let M be a complete manifold without boundary. Suppose q is a C z function defined on M. For any 3.>0, we consider H~(x, y, t), which is the funda
mental solution of the equation
( A  ~ 2 q  ~ t t ) u(x, t)=O
on MX(O, ~). Then
lim log Hx(x, y, t/g)
~. where O(x, Y, t) is defined by (3.3).
2
=  O ( x , y , t)
ON THE PARABOLIC KERNEL OF THE SCHR(~DINGER OPERATOR
197
Proof. For any given x, y E M, let Bp(R) be a geodesic ball with radius R containing x and y. Applying Theorem 2.1 to Ha(x, y, t), we have Ha(x'x' tl)<'Ha(x'Y' t2) \~1]
where
(t2~ hal2
exp(Aa(t2tl)+Qa'R;a(x'Y' t2t'))
Aa = C7[a R  1w r g +
ot3(a _ 1) 
In 2 +22/3 y2/3(a_ 1) v3a  ~/3+ (aO)1/22+ a ( a  1) IK]
with K, O, and ~, as defined in Theorem 2.1. Also Qa.R,a(x, y, t) is the metric defined by (3.1) with 22q replacing q. Taking log of both sides, we have log HA(x, x, tl/2) logHa(x, Y, t2/2) na ~< 2 +~l~ 2 Letting 2+ ~, and observing that
t2 + Aa(t2tl)
22 ~
Qa,R;a(x, Y, (t2tl)/2)
2
Qa.R;a(x, y, t/2) = Qa,R(x, y, t) 2
and lim Aa a, ~ = 0, we conclude that lim log Ha(x, Y, t2/2) t> lim l~
2
x, ti/2)
2
a~
2*=
~a,R(x, y, t2tl).
(6.1)
Letting R+oo, and a~l, this gives lim a,oo We now claim that lim lim
tl~0 a~0o
log Ha(x, x, tl/2) log HA(x, y, t2/2) I> lim 2 a,= 2
p(x, y, t2tl).
(6.2)
log HA(x, x, tl/2)
2
I> 0,
and the lower bound will follow. Indeed, if qo>q on Bp(2R), then the kernel cq~ y, t) satisfies the equation
198
PETER LI AND SHING TUNG YAU
with l:l(x, y, t) being the heat kernel with Dirichlet boundary condition on Bp(2R). Hence, by the assumption qo>~q and the maximum principle,
H~(x, y, t) ~ e
on Bp(2R) XBp(2R) x [0, oo). In particular log Ha(x, x, tl/2 )
n[x, y, t)
I> qo t~§
log/at(x, x, t l/2 )
(6.3)
However, by the asymptotic formula for l=l(x, x, t) as t~0, lim ,t, log fl(x, x, h/2)
= 4, ~ \
I ( l o g H ( x , x , tl/g.)
~
= O.
Hence, after letting ,~~, (6.3) becomes lim logHx(x, x, tl/2)
>~qo tl,
and the claim follows by letting h~0. To establish the upper bound, we employ Theorem 3.3, which gives
log H~(x, y, t/g )
2
~< 2 I log (1 +6) n V1/2(SI) VIn(S2)+22Ax 6(2+6) t
+Oa, s(x, $2,6(1+6) t)+Oa, R(y, S I, 60 + O(x, $2,6(1+6) t)_O(x, S~, (1+26) (1+6) t).
Letting 2.0o, then a~l and R~oo, we get lim log Ha(x, y, t/A) ~<20(x, S2, 6(1 + 6) t)+O(Y, S I , 6t)_Q(x, S 1, (1 +26) (1 +6) t). ~, 2
Setting S2{x ) and S~={y}, we derive the inequality
(6.4)
lim ~,
log Ha(x, y, t/2 ) <~20(x, x, 6(1 +6) t)+Oty, y, 6t)O(x, y, (1 +26) (I + 6) t). (6.5) 2
On the other hand, taking F: [0, 1]~M to be the trivial curve with y(s)=x, we see that
O(x, x, t) <~tq(x).
Hence, by letting 6~0 in (6.5), the upper bound follows.
ONTHEPARABOLIC KERNELOF THESCHRODINGER OPERATOR
199
Remark. Since the techniques used in the above theorem are completely local in
nature, Theorem 6.1 is still valid when the manifold is compact with or without boundary, for any boundary condition. In the case when OM*f~, we need a version of the Harnack inequality which is valid for any points which are e distance away from aM. Such a Harnack inequality can be derived using the method employed in w1 and w2. In that case, the estimate will depend on e and the geometry of aM.
Appendix
We will establish the fact that the function O(x, y, t) defined in w2 is Lipschitz for a locally hounded potential function q. Let r be the geodesic distance between y and z in M and Y2 be a geodesic joining y to z which realizes distance. For any e>O, we can find a curve Yl parametrized by s E[O, 1] joining x to y such that
O(x, y,t)+e >~l fo' ly,, ds+t fo' q(F,) ds.
We define a new curve 7 by y(s) = IY, ( i ~ r ) , ~72(s+r1), Clearly ~ s ) is a curve joining x to z. Hence if0~<s< l  r if 1r<~s<~ 1.
 1 O(x, Z, t) <~4t yO 1~'12ds + t fOI q(y(s) ) ds
<1_ 4t
fo,ri~ql2(l_r)_2 ds+
r
1~,212 ds+t
qOq(s)) ds+trqo '
where q0 is the supremum of Iql in a neighborhood containing the curve y. By a change of variable, the above inequality yields
O(x, Z, t) <~ 4t(lr~f01 ]Fl[2ds+ + t ( 1  r ) l ~t = l_r(O(x,y,t)+e)+
Hence
r O(x,z,t)_O(x,y,t)<~_~_rO(X,y,t)+
f01q(?l(s)) ds+trq o
t
l
( l~r_ r )  
f01q(?l(s))ds +r ~+trqo.
1r
1r
e
r tqo+ r ~"
200 Letting e>0, and using
PETER LI AND SH1NGTUNG YAU
Q(x, y, t) <~ r2(x' y) +tq o, 4t we conclude that if r(y, z)~<89then
O(x, z, t)O(x, y, t) <. r(y, z) "C,
where the constant C d e p e n d s on qo, t, and r(x, y). Reversing the role of y and z yields the desired Lipschitz p r o p e r t y o f O(x, y, t).
References
[1] ARONSON, D. G., Uniqueness of positive weak solutions of second order parabolic equations. Ann. Polon. Math., 16 (1965), 285303. [2] AZENCOTT,R., Behavior of diffusion semigroups at infinity. Bull. Soc. Math. France, 102 (1974), 193240. [3] CHEEGER,J. ~: EBIN, D., Comparison theorems in Riemannian geometry. NorthHolland Math. Library (1975). [4] CHEEGER, J. ~; GROMOLL, D., The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geom., 6 (1971), 119128. [5] CHEEGER, J., GROMOV, M. & TAYLOR, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom., 17 (1983), 1533. [6] CHEEGER,J. & YAU, S. T., A lower bound for the heat kernel. Comm. Pure Appl. Math., 34 (1981), 465480. [7] CHENG, S. Y., LI, P. & YAU, S. T., On the upper estimate of the heat kernel of a complete Riemannian manifold. Amer. J. Math., 103 (1981), 10211063. [8] CHENG, S. Y. & YAU, S. T., Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math., 28 (1975), 333354. [9] DONNELLY, H. & LI, P., Lower bounds for the eigenvalues of Riemannian manifolds. Michigan Math. J., 29 (1982), 149161. [10] FISCHERCOLBRIE,D. & SCHOEN,R., The structure of complete stable minimal surfaces in 3manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math., 33 (1980), 199211. [11] FRIEDMAN,A., On the uniqueness of the Cauchy problem for parabolic equations. Amer. J. Math., 81 (1959), 503511. [12] GALLOT,S. & MEYER, D., Optrateur de courbure et Laplacien des formes difftrentielles d'une varitt6 Riemannienne. J. Math. Pures Appl., 54 (1975), 259284. [13] GROMOV, M., Paul Levy's isoperimetric inequality. IHES preprint. [14] w Curvature, diameter, and Betti numbers. Comment Math. Helv., 56 (1981), 179197. [15] KARP, L., & LI, P., The heat equation on complete Riemannian manifolds. Preprint. [16] LI, P. & YAU, S. T., Estimates of eigenvalues of a compact Riemannian manifold. Proc. Sympos. Pure Math., 36 (1980), 205239. [17] MAUREY,B. & MEYER, D., Un lemma de gtomttrie Hilbertienne et des applications h la gtomttrie Riemannienne. Preprint.
ON THE PARABOLIC KERNEL OF THE SCHRODINGER OPERATOR
201
[18] MEYER,
[19] [20] [21] [22] [23] [24] [25] [26] [27]
D., Un lemme de g6om6trie Hilbertienne et des applications a la g6om~trie Riemannienne. C.R. Acad. Sci. Paris, 295 (1982), 467469.   Sur les hypersuffaces minimales des vari6t6s Riemanniennes a courbure de Ricci positive ou nuUe. Bull. Soc. Math. France, 111 (1983), 359366. MOSER, J., A Harnack inequality for parabolic equations. Comm. Pure Appl. Math., 17 (1964), 101134. SERRIN, J. B., A uniqueness theorem for the parabolic equation ut=a(x)u~x+ b(x)ux+c(x)u. Bull. Amer. Math. Soc., 60 (1954), 344. SIMON, B., Instantons, double wells and large deviations. Bull. Amer. Math. Soc., 8 (1983), 323326. VAROPOULOS,N., The Poisson kernel on positively curved manifolds. J. Funct. Anal., 44 (1981), 359380.   Green's functions on positively curved manifolds. J. Funct. Anal., 45 (1982), 109118. WINDER, D. V., Positive temperature on the infinite rod. Trans. Amer. Math. Soc., 55 (1944), 8595. YAU, S. T., Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math., 28 (1975), 201228.   Some functiontheoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J., 25 (1976), 659670.
Received June 7, 1984 Received in revised form February 14, 1985
14868283 Acta Mathematica 156. Imprim~ le 15 mai 1986