Movement of hot spots in Riemannian manifolds

M O V E M E N T O F H O T S P O T S

I N R I E M A N N I A N M A N I F O L D S

By

ISAAC CHAVEL t AND LEON KARP tt

Let (M n, ds 2) be an n-dimensional Riemannian manifold, dVthe Riemannian measure, and p(x, y, t) the minimal positive heat kernel, of(M", ds2). Then for

any ~0 >_- 0 in L~ (M),

(Pt~a)(x) = " f p(x ,y , t )qJ(y)dV(y)

M

is the minimal positive solution of the heat equation on (M ~, ds 2) satisfying

lim Pt~a = g tlo

at all points of continuity of ~0. Thus, for M" -- R" with canonical Riemannian

metric, we have

= f (47tt)-n/2e-Ix-rl2/4t~7(y)dy (P/p)(x)

R"

where dy is the usual Lebesgue measure. (See [3, Chapters VI-VIII].) Our primary interest is in Riemannian manifolds where for every nonnegative

~o in L~ ~ (M), the locus

H(t) = H(t; g)=" ~x " (Pt~)(x)= max (Pt{o)(Y)[ t y J

of "hot spots" is compact for all t > 0. The collection of such Riemannian manifolds is reasonably large. Besides the compact ones, the class contains those noncompact Riemannian manifolds for which for all t > 0, r EL~(M), we have (PtqO(x) ~ 0 as x --- oo (see [ 12], [5], [7] for results on the geometric hypotheses guaranteeing this condition). Here we initiate a study of the location of H(t) relative to the support of ~a, S,, and, more precisely, the behavior of H(t) a s t t +oo.

t Supported in part by NSF grant DMS 8704325 and PSC-CUNY FRAP awards. tt Supported in part by NSF grant DMS 8506636 and PSC-CUNY FRAP awards.

271 JOURNAL D'ANALYSE MATH~,IATIQUE, Vol. 55 (1990)

272 I. CHAVEL A N D L. KARP

We emphasize that this is not the study of the maximum values of Pt~. Rather,

our interest is actually in the function

O, = Pt~o/max Ptg,

which we refer to as the profile ofPt~o. Thus our study of H( t ) as t t + oo reduces to the study of {~t -- 1 } as t t + oo. If ~t ~ ~ everywhere on M, as t t + oo, for

some function ~ , then we refer to �9 as the limiting profile ofPt~o. Of course, might very well exist, and not even have maximum value equal to 1. However, in those cases where we know in advance that there exists a compact K containing

H(t) for all t > 0, and ( I ) t ~ (I) as t t + 0% then the limit points of H(t), as t t + oo, are located in {~ = 1 }.

Our main results here are as follows:

(1) If M is Euclidean space R", hyperbolic space H", or the sphere S", of dimension n > 1 with standard Riemannian metric, then H(t) is always con-

tained in the closed convex hull, C~, of the support of ~0, S~. (Of course, for the

sphere S", if S~ cannot be contained in any closed hemisphere then C~ = S".) (2) For M" --- R", let m~ denote the Euclidean center of mass of 9- Then

H(t) ---, (m~}

a s t t + ~ . (3) For hyperbolic space, there exists a single point h~ for which

H(t) ~ {h~}

if S, is sufficiently small. Otherwise, one gives an easy counterexample to H(t) converging to a single point.

(4) I f M = S" then we have a special case of the study of H(oo) (the limit set of

H(t) as t t + o0) in general compact Riemannian manifolds, the analysis of which is obtained from the Sturm-Liouville decomposition of L 2. For M = S" the result

is: if 9 is not L2(S")-orthogonal to the restriction of the coordinate functions of

R" +' to S", then H(oo) consists of one point. Of course, since S" is compact, Pt9 has a locus o f"co ld spots". In the case under consideration, the limiting position

of the "cold spots" as t t + oo is - H(oo). (5) One is tempted to believe that the general result would be: Given noncom-

pact (M", ds 2) for which Pt~o '-" 0 as x ~ oo for all t > 0 and compactly supported

q~, then there exists a compact set Kin Msuch that H(t) c_ Kfor all t > 0. We give a rather elementary counterexample to this proposed result. So to extend the

theory beyond the classical spaces requires geometric hypotheses on the spaces in question. Even in those cases where H(t) is not uniformly bounded, one might

ask for an upper bound on the rate at which "hot spots" escape to o0. We note that in our counterexample, which has constant curvature equal to - 1, the rate at

MOVEMENT OF HOT SPOTS IN RIEMANNIAN MANIFOLDS 273

which H(t) escapes to ~ is essentially equal to t; however, when M is Rieman- nian complete with nonnegative Ricci curvature, then for any x ~H( t ) we have

(I) d(x, S~) < const, x / : /~ t

for sufficiently large t. The ~ t may be improved to v/t if we add the assumption of "maximal volume growth", i.e., that

V(x0; r) > const, r n

as r t + oo, where V(xo; r) denotes the volume of the metric disk of radius r

centered at some fixed x0 in M. However, it seems reasonable to conjecture that, for manifolds of nonnegative curvature, H(t) is contained in a fixed compact set

for all positive t. We note that one might wish to view our results as "inverse results" relating

(especially in R n) the asymptotics of Pt~o to the geometry of ~. In this spirit one

might ask an even stronger question, viz., to what extent does the asymptotic behavior of Pt~0 determine the initial data ~o?

As a preamble to our result concerning this question, recall the open question

of the uniqueness of gravitational potentials: If f~l and f~2 are two solids in R 3 (a solid is a compact set equal to the closure of its interior, for which the

complement of its closure is connected), and their gravitational potentials

Pota(x) = Ix -- y I

agree on some open subset of (and hence throughout) the unbounded component o f R 3 \ (~, U ~2), can one conclude that ~ --- ~2? It is known [1], [13] that the

answer is negative in general for arbitrary solids f~l, ~2, and positive if one of the solids is a 3-disk in R 3 (the "Potato Kugel Theorem" of [ 1 ]). The question remains

open if it is only assumed that one of the solids is convex (see the discussion in [13], and the references therein). If we consider the heat equation in place of

Laplace's equation, we are led to ask: if ~0~ = IB, the indicator function of an n-disk in R n, and ~2 = In, ~ an arbitrary relatively compact open subset of R ~,

does the "agreement"of PJa with PtIB imply that indeed ~ -- B (a "Hot Potato Kugel Theorem")? In fact we have

(6) IfPtla agree with PtIB on an open set Uup to order t -(n/2+l) a s t ~ -t- oo, then

~ = B, modulo a set of measure 0. (If, instead, we assume ~ is a solid, then n--B.)

(7) I f ~ is a solid, and the level sets of Ptla in R" remain invariant for all t > 0, then ~ is an n-disk.

It is natural to ask whether ihe result (6) remains valid for ~ , ~2, one of which is assumed to be convex.

274 I. CHAVEL AND L. KARP

1. E u c l i d e a n space

We are given r > 0 in L~(R"). Then, as mentioned earlier,

(2) (Pt~)(x) = : ( 4 nt ) - n/2 e - l X - r l2/4t ~ (y )dy .

R ~

Recall that m~ denotes the center of mass of ~, S r the support of ~, C r denotes

the closed convex hull of S r, and H ( t ) the locus of maxima points of Pt~.

T h e o r e m 1. For each t > O, the function Pt~ has a m a x i m u m value, and it

can only be at tained at points o f C r, i.e.,

(3)

Furthermore,

(4)

as t t + oo.

Proof .

H ( t ) C_ C r.

H ( t ) ~ {mr}

To each x ~ R n \ C r there is a unique nx EOCr for which

d(x , Cr) = Ix - nxl.

For any y in the interior of the support of ~, the function

~ I ~ - Y l

decreases as ~ moves from x to n~ along the line segment Xnx. By (2), Ptq~ increases as ~ moves from x to nx along the line segment xn~. This implies (3).

For (4) we note that (2) implies

(5) (VPt~)(x) = - ( a n t ) - , , 2 : (x ~ y) e - Ix-yl2/4t~(y)dy

so that for every maximum point h t o f et~ we have

f (ht -y)e-Ih,-Yl2/4'~(y)dy = 0. (6)

Let (tk) be a sequence converging to + ~ as k t + ~ ; since all the points htk are

confined to a compact set, (htk) possesses a subsequence (hT,) which converges to some point h~ in C r. But then

, f (h~o - y)q~(y)dy = O,

i.e., h~ = m r, the center of mass of tp. Therefore, the center of mass is the only allowable limit point for subsequences (h 0 of the original (ht~), which implies the original (htk) converges to the center of mass of ~. But since the sequence (ht~) is


arbitrary, we have (4) as t t + oo. []

R e m a r k 1. We note here that one can easily estimate the rate at which

H(t) ~ {mr}, viz., we claim that for ht EH( t ) we have

I h, - m~,l < (diam S,) 3 (1 + O(1/t)) 4t

as t ~ + oo. Indeed,

I ht - m, l e-~'am c,'2:4t : q~(y )dy < l ht - mr l l f e - lh,-;':/4t~o(y )dy [

= l J" (m, -- y)(1-- e-lh,-yl2/4t)~o(y)dy I

< f I m r - Y l I h t - Y l 2 = 4t q~(y)dy

< (diam Cr) 3 : = 4t ~o(y)dy

m the equality a consequence of (6). Thus,

I ht - m~, l < (diam Cr) 3 etai~mc, w4t. 4t

We leave

follows.

T h e o r e m 2. The level sets of Pt~ tend to spheres centered at m r as t ~ + oo.

Proof . We have, from (5),

2t(VPt~o)(x)(4rtt) "/2 -- - f (x -y)e-Ix-yl2/4t~(y)dy J

it to the reader to check that d iam C r - - d i a m St, and the claim []

as t ~ + oo. Thus the limiting position of the gradient vector field of VPtr is a

constant multiple of the position vector field x - rag, with origin at m, . This

implies the claim. []

We say that the levelsets of Pt~ are invariant in time i f any locus L in R n is a level

set of Pt~ for a t ime t = t, i f and only i f L is a level set of Pt~ for all t > O. (We are

= -- : (x -- mr)e-Ix-y]214t~(y)dy

-- : (m 9 -- y)e-bx-Yl'/4tq~(y)dy

-" -- (x -- m,) : ~(y)dy


not requiring that the value Pt~o ] L be invariant with respect to t ime - - only that

Pgo ] L always be a constant, for each t.) We may conclude that

C o r o l l a r y 1. Pt~o has level sets invariant in time if and only if S, is rotation invariant about its center of mass.

P r o o f . Since the level sets tend to spheres about m, , and are invariant in

time, it follows that the level sets o f S, (i.e., at t ime t = 0) are spheres centered at

m 9 . []

R e m a r k 2. We refer to [8], [13] for their discussions o f a different problem

regarding t ime invariant level sets o f solutions to the heat equation, and to its

solution in [2].

We now consider ~0" the (Schwarz) symmetrizat ion o f ~o, and the associated

asymptotic expansion of

t"/E{e,(~0 -- q~*)}(x)

as t t + ~ . Recall that q,* is a nonnegative function defined on the closed disk K*

possessing the same volume as K = : support ~0. For convenience, we center the

disk K* at m~. To each a we set K, to be the set of points in R" on which q7 > a, and

K* the closed disk, centered at m, , whose volume is equal to that o f K~. Then q,* is

defined by

(p*(x) = : sup{a :xEK*} .

Note that for ~p = IK, the indicator function of K, ~o*= Ix.. Furthermore, for

arbitrary ~0 we have

(7) f ~ ( y ) d y = f ~*(y)dy.

T h e o r e m 3. We have

'y (8) (4~t)"/E{Pt(~o - ~o*)}(x) ~ - t4" lY 12(~ o - ~*)O')dy

as t t + oo. I f ~o = Ix, then

(9) l i m tn/2+let(~o - ~o*) ~ 0 t t + ~

with equality i f and only if K is a disk (modulo a set of measure O)

P r o o f . We have


(4m)n/E(Pt(~o -- ~o*)}(x)

=~, (-4t)-~ f k=0 k! Ix - y 12k(r -- q~*)(y)dy

y ly = ( e - ~ o * ) O , ) d y - ~ I x - y I~(~o - ~*)O,)dy + O(t-~)

I f ---4t { I x l 2 - 2 ( x ' Y ) + lYl2}(~~

= - 4t lY 12((P - ~a*)(y)dy + O(t-2),

where we have used the fact that m r is also the center of mass of (p* (since K* is

centered at mr); this implies (8). For (9) we need only note that

f l Y - ~ 0 12(I~ I ,,. ) ty )dy

with equality if and only if K is a disk (modulo a set of measure 0). []

A slight modification of the above proof yields

T h e o r e m 4. (The "Hot Potato Kugel Theorem") Given ~ a relatively

compact open set (resp., a solid), and an n-disk B in R n, and assume that the limit

L ( x ) = : lim tm2+lPt(I n - In)(x) t t + oo

exists and is identically equal to O for x in some open set U contained in the interior

o f R n \ {B U ~}. Then f l = B up to a set o f measure 0 (resp. ~ = B).

P r o o f . From the expansion of

t "'2+ 'P,(I, -- IB)(x)

as above, we have (i) from the existence of the limit L ( x ) that V(B) = V(•), and, therefore, (ii)

0 = L ( x ) = const, f { Ix 12 - 2(x, y ) + lY 12)(/n - Ia }(y )dy

on U. Now differentiate with respect to x. Then

f y(In - IB)(y)dy = O,

so that fl and B have the same center of mass. The result now follows from

Theorem 3. []


2 . H y p e r b o l i c s p a c e

(For general background we refer the reader to [3, chapters VI, X, XI], [6, pp. 116-122].) We now consider H n, with distance function d(x,y) . Then the solution to the initial value problem is given by

= f p(x, y, t)97(v)dV(y), (Pt97 )(x ) H n

where p(x, y, t) denotes the heat kernel of H n, and dV is the Riemannian

measure. For H ~, the heat kernel p(x, y, t) depends only on d(x, y) and t; so we have

p(x, y, t) = ~(d(x , y), t).

It is well known that for each t > 0, $'(r, t) decreases as r increases. Again, for any 97 >= 0, let C 9 denote the closed convex hull of the support of 97

(for any set E, the closed convex hull of E consists of the intersection of all closed (hyperbolic) half-spaces containing E); and for any x ~ H" \ C~ one has a unique

nx E OC v for which

d(x, G ) = d(x,

which, in turn, determines the geodesic ?x connecting x to a~. Now one uses the first variation (of arc length) formula [9, p. 80] to show that, for any y E C v, d({, y) decreases as ~ moves along 7, from x to 2,, which implies

T h e o r e m 5. In H" we have H(t) c_ C~for all t > 0, i.e., the maximum points of Pt97 are located in C~, for each t > O.

R e m a r k 3. We note that the proof of Theorem 5 easily carries over to the spheres S ". For the study of large-time behaviour of H(t) for the spheres S n, see Section 4 below.

We also note that the above argument may be carried out on any two-point homogeneous space, since the heat kernel in those spaces depends only on time and on distances between points, and decreases with respect to distance (see [4], [3, pp. 1 92-1 94]). One requires, however, a suitable notion of convexity in those spaces. We omit the details here.

To study what happens when t t + oo, we let, for any x ~ H n, exp~ : (H")~ ~ H ~

denote the exponential map of the tangent space o f H n at x, (H~)x, to H". For each x E H ~, exp~ is a diffeomorphism. Then for

( P t 9 7 ) ( x ) = f ~(d(x , y), t)97(y)dV(y),

we have


(10)

SO

(ll)

f exp,- ~y 0 (VPt~~ = -- [exp~-~y [ Or - - (d(x , y) , t)~o(y)dVty).

To carry out an explicit calculation, we consider the case n = 3. Then

e - t r ~ ( r , t ) ---- - - - - e - r 2 / 4 t ;

(4nt) m sinh r

O , e - te -r'/4t { rcosh r r~t } Orr (r, t) = (4nt)3/2 sinh r sinh r

Therefore, for t ~ + ~ , we are led to study the limiting profile of ~ ( r , t),

f(r) = r /s inh r,

with limiting profile of (O~/Or)(r, t),

d r sinh r - r cosh r f ' ( r ) = - - =

dr dr sinh r sinh 2 r

One verifies that

f ' < 0 V r > 0; f " (0) < 0; (12)

f " > O V r > 2 .

So there exists a maximal roE(O, 2) such that

(13) f " < 0 Vr ~[0, ro).

Given q~, the limiting profile ofPtt, o, F, is given by

(14) F ( x ) = f f ( d ( x , y))~o(y)dV(y).

From (11), (12) one shows that if the support of ~o is contained in an open disk D of radius ro/2, then the Hessian of F on D is strictly negative. There- fore, F has precisely one maximum. So there exists a single point h~o for which

if S~ is sufficiently small.

H ( t ) ---. {h~}

However, if the support of ~o is sufficiently large, then we lose the uniqueness of the limiting positions of the hot spots. Indeed, let

280 I. C H A V E L A N D L. K A R P

where x I ~ x2, i.e., the initial data consists of two point sources. Then the limiting profile of Ptr F, is given by

d(x, xl) d(x, x2) F(x) = -~ sinh d(x, xO sinh d(x, x2)

If F had a unique maximum, it would have to be the midpoint Xo of the geodesic segment connecting x~ and x2. But for x~, x2 sufficiently far apart, it is clear that

F(x,) = F(x2) >= 1 > F(xo).

A different version of this example, with r = IK, is the dumbbell, which is constructed as follows: Fix R > 0, and a ~> 1 ~> b. Let 7(s) be a unit-speed geodesic in H 3, Xo = ~(0), x~ = ~,(a + R), x2 = ; ' ( - (a + R)). Let Ta, b denote the tubular neighborhood ofT([ - a, a]) of radius b, and set K = Ta.b U B(X~; R) t3 B(x2; R).

Consider PtP for q~ = IK. Again, if F(x) has a unique maximum, it must be at 7(0). Then F(~,(0)) is estimated by

F(7(O)) = f d(~,(0), y) sinh d(~,(0), y)

K

dVO')

b a

(s inh b~ <-_4rcfrsinhrdr+4nfrsinhr{1-cosarcsinksi---n-~r/}dr

0 b

+ 2 f d(y(0),y) sinh d(7(0), y)

B(xl; R)

dV(y)

2b a

_-< 4~ f r sinh rdr + 4rt 0 2b

f rs inhr{1- cos arcsin (s inh b ~ ~ dr k sinh r / J

+2 d(xo, y)

f sinh d(xo, y) B(xg R)

dV(y)


2b a

(sinh b ) < 4zt f rsinh rdr + 4rr y r(sinh r )arcsin \ s~nhr/dr

0 2b

2 a + ~ V(xl; R)

sinh a

a

I dr < const, b 2 + 4zt ~ r sinh b + z a

d / sinh 2 b sinh a 2b 1

sinh 2 r

V ( x I " ~ R )

a

< const.{b 2 + ba 2} + const.R sinh a"

In summary, a

F(7(0)) --<_ const.{b 2 + ba 2} + const.~ sinh a"

We shall first fix R, then a, then b. Certainly, once R is fixed, we can choose a

sufficiently large and then b sufficiently small to guarantee that F(7(O)) is as small

as we like. On the other hand, R

F(xl) > 4zr f r sinh rdr ;

0

we may therefore fix R, a, b, to guarantee that 7(0) is not the maximum

o f F .

3. A hot circle migrat ing to infinity

We consider the surface of revolution M 2

given by

d s 2 = d r 2 + e2rdO 2,

with Riemannian metric

where - oo < r < + 0% 0 ~ S ~. Then the associated Laplace operator is

given by

0 z 0 82 - 2 "

002

For functions ~o depending only on r, the Laplacian Ar has constant coefficients, so the initial value problem for the heat equation with initial data given by ~0 = ~0(r) may be solved using the Fourier transform. One obtains


+oO

i f (Ptr = ~ e-(r-'+~162 "V

- - o 0

So for ~ = ~po, the delta distribution on the line concentrated at p = P0, the maximum of P~p is located at ht --Po - t. More generally, if q~ had support [ - A, A ], then

ht < - t +A. So h,--" - ~ a s t t + ~ .

4. T h e sphere

Let M denote a compact Riemannian manifold, with Sturm-Liouville decomposition:

{0----/],.0 < ~1 __---~22" �9 " ~ § OC)},

{~,0, ~,,, ~ , . . . },

where 2 2 denotes the j- th eigenvalue (counting multiplicities) of the Laplace operator A, with associated normalized eigenfunction %, such that the collection of eigenfunctions form a complete orthonormal sequence in L2(M). Given tp ~L| the solution to the initial-value problem for the heat equation, with initial data ~, is given by

(15)

where

(16)

(Pgo)(x) = i aje-a/Ctj(x), j - O

aj = f ~oyjdV

M

and dVdenotes the Riemannian measure on M. Since ~'o(X) = 1 / ~ , where V(M) is the volume of M, we have

which implies

1

M

( e , r - - 1 f dv, V(M) j -

M

Of course,


1 : lim Pgo = ~odV.

t ~ + | V ( M ) M

It'is well known that each distinct eigenvalue has finite multiplicity. We enumer-

ate the distinct eigenvalues by {0 = 2o < 2~ < 22 < �9 �9 �9 t + oo } with their respect-

ive eigenspaces {Eo, El, E2 . . . . }. (See [3, Chapters I, VI].)

Proposition 1. The limit

lime1't{Pgo i f v} t t + ~o V ( M ) ~od

M

is given by the projection of q~ onto the eigenspace Ek, where ]~k is the lowest positive eigenvalue for which ~o has a nonzero projection onto the associated eigenspace.

Proof. Let ~k = •l . . . . . 2 1 + h - 1. Then al . . . . . a t - i = 0 and

f Ptq~ V(M) j-l M

l + h - I ~, =e -~: ~ aAuj+

j = l j = l + h

aje - 2lt~ff j ,

with 2 i > 2k for all j >= l + h. We conclude

lim e~t{pt~ 0 1 f V} t+h-~ tt + ~ V ( M ) qM = J=JY" aj~,j ,

M

which is the claim. []

Remark 4. See [ 1 1 ] for a slightly restricted version of the above argument.

Theorem 6. Let M --- S" ___ R "+1, x = (x~ . . . . . X,+l) the position vector in R " + ~ , r

f ~odV § O,

M

then all maximum points of Pica tend to the spherical "center of mass"

m,=: f odV/ [ f odV, M M

and all minimum points of Pgo tend to - m, as t t + ~ .

284 I. CHAVEL A N D L. KARP

Pr oof . The whole issue is that for M = S", the eigenspace E~ is spanned by the functions

(~,/x/(n + I)V(M) . . . . . ~n+l/x/(n + 1)V(M)}.

Thus the hypothesis is, in fact, that the projection of ~0 onto E~ is nonzero. Now use the previous argument. []

R e m a r k 5. If the projection of ~ onto El, for M = S ~, is 0, then one has different possibilities. For example, if ~ = Iu and f~ is an equatorial

band centered on the equator, one can easily calculate that H(t)--, equator as t t + oo, and the cold spots migrate to the poles. Naturally, if the band is shifted by e > 0 toward one of the poles P, then H( t ) - , P and the cold spots tend to

- P (since its projection onto El is nonempty and one can explicitly calculate m r

to be P).

5. N o n c o m p a c t m a n i f o l d s o f n o n n e g a t i v e R i c c i curvature

We let M be complete noncompact with nonnegative Ricci curvature. Then

one has (i) [10, Theorem 2.2(i)] for any positive solution u(x, t) to the heat equation on M, that

In/2u(x, t)

is an increasing function with respect to t, and (ii) [10, Corollary 3.1] the estimate

p(x, y, t) < C(e) e - d~x, yV(4+ ~),

on all o f M • M • (0, + oo), for any given eE(O, 1). We have

u(x, t) = 3 p(x, y, t)~o(y)dV(y).

M

Fix x0 in S~, ~ ~ (0, 1), and set

a =" ~"'2U(Xo, ,5).

Then for h t ~ H ( t ) , and t _-__ 1, we have


a ___< U(Xo, t)t "'2

<= u(h. t)t "a

= t "'2 f p(ht, y, t)qJ(y)dV(y)

M

t n / 2 v ( s ~ ) c ( g ) II II sup y~s,

e - d~h,, y ~(4 + ~ )t

v y;

const , t"/zv(g~)f(e) II II sup yESp

e-d2(hs, yy(4+e)t

V ( xo ; ,fit)

which implies

inf - - d Z ( h " y) <-_ In (t"'zv(s~)(2(e) II ~o II ~&V(xo; xfi)} ms, (4 + e)t

which implies

as t ~ + ~ , in general , a n d

i f

d(h, S~) < const, x / t In t

d(h. S~) < cons t , x / t

V(xo; r) inf - - > 0. R>0 r n

ACKNOWLEDGMENT

It is a pleasure to thank E. A. Feldman and S. Elizur for helpful and stimulating

conversations.

REFERENCES

1. D. Aharonov, M. M. Schiffer and L. Zalcman, Potato kugel, Isr. J. Math. 40 (1981), 331-339. 2. G. Alessandrini, Matzoh ball soup: A symmetry result for the heat equation, J. Analyse Math.

54 (1990), 229-236. 3. I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984. 4. J. Cheeger and S. T. Yau, A lower bound for the heat kernel, Commun. Pure Appl. Math. 34

(1981), 465-480. 5. J. Dodziuk, Maximum principleforparabolic inequalities andheat on open manifolds, Indiana

Univ. Math. J. 32 (1983), 703-716. 6. D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and

measured pleated Riemann surfaces, in Analytic and Geometric Aspects of Hyperbolic Space, D. B. A. Epstein, ed., Cambridge University Press, Cambridge, 1987, pp. 113-253.

7. P. Hsu, Heat semigroup on a complete Riemannian manifold, preprint. 8. M. S. Klamkin, A physical characterization of a sphere (Problem 64-5"), SIAM Rev.

6 (1964), 61.

286 i. CHAVEL AND L. KARP

9. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. H, Interscience Publishers, New York, 1969.

10. P. Li and S. T. Yau, On the parabolic kernel of the SchrOdinger operator, Acta Math. 156 (1986), 153-201.

I 1. J. Rauch, An introduction to the qualitative theory of partial differential equations, in Partial Differential Equations andRelated Topics, Lecture Notes in Math. 446, Springer-Vedag, Berlin, 1975, pp. 355-369.

12. S. T. Yau, On the heat kernel of a complete Riemannian manifold, J. Math. Pures Appl. 57 (1978), 191-201.

13. L. Zalcman, Some inverse problems of potential theory, Contemporary Math. 63, Providence, R.I., Am. Math. Soc., 1987, pp. 337-350.

Isaac Chavel DEPARTMENT OF MATHEMATICS

THE CITY COLLEGE OF THE CITY UNIVERSITY OF NEW YORK

NEW YORK, NY 10031, USA

Leon Karp DEPARTMENT OF MATHEMATICS

LEHMAN COLLEGE OF THE CITY UNIVERSITY OF NEW YORK

BRONX, NY 10468, USA

(Received October 23, 1989)

Movement of hot spots in Riemannian manifolds

Documents

Transcript of Movement of hot spots in Riemannian manifolds