A MATRIX HARNACK ESTIMATE FOR THE HEAT EQUATION RICHARD S. HAMILTON

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1 COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 1, Number 1, , 1993 A MATRIX HARNACK ESTIMATE FOR THE HEAT EQUATION RICHARD S. HAMILTON In the important paper[ly] by Peter Li and S.-T. Yau, they show how the classical Harnack principle or the heat equation on a maniold can be derived rom a dierential inequality. In particular, they show that or any positive solution / > 0 o the heat equation dt on a compact Riemannian maniold o dimension m solving the equation or t > 0, i the maniold has weakly positive Ricci curvature ijy > 0 then or any vector ield V on M J?t + L + 2D(V) + \V\*>0 ] a similar result holds with an error term i the Ricci curvature is bounded below. The quadratic version in V given here is equivalent to the more complicated ormula in their paper by choosing the optimal V. We shall show in this paper that the Harnack estimate o Li and Yau is the trace o a ull matrix inequality. Main Theorem. I M is a compact Riemannian maniold and > 0 is a positive solution to the heat equation on M dt or t > 0, then or any vector ield V* on M we have DiDj + ^ ga + A/ ^ + 2V K + V^ > 0 J Reseaxch partially supported by NSF contract DMS

2 114 RICHARD S. HAMILTON provided M is Ricci parallel and has weakly positive sectional curvatures; while in general we can ind constants B and C depending only on the geometry o M (in particular the diameter, the volume, the curvature and the covariant derivative o the Ricci curvature) such that m / 2 / < B and or all Vi DiDj + 1 gy + DJ V s + Dj ^ + V^ + C/[l + log (B/r /2 /)]</«> 0. This inequality is equivalent to Theorem 4.3 by choosing the optimal Vi = Di/. The result can also be proved or bounded positive solutions on complete maniolds with bounded curvature and derivatives (we omit this technical detail). One can easily check that the inequality becomes an equality or the undamental solution on Euclidean space R m - J_ P -\x\ 2 /4t J t m/2 e which is a homothetically expanding soliton along the vector ield Vi = Xi/2t. The hypothesis that M is Ricci lat with weakly positive sectional curvatures is satisied on a torus or a sphere or a complex projective space, or a product o such, or a quotient o a product by a inite group o isometries. In this case the Harnack estimate becomes which is a kind o logarithmic convexity. In particular, along any geodesic x(s) para-metrized by arc length 6 we have and hence s 2 lo g + - t is convex; rom which we conclude that along any geodesic x(s) (x(0),t) < (x(s),t)^(x(-s),ty^^. Also i / has its maximum along the geodesic x(s) at x = 0 at time t then everywhere else on the geodesic (x(s),t)>(x(0),t)e -s2 /4t

3 A MATRIX HARNACK ESTIMATE 115 Similar results with small error terms can be derived or maniolds M with general curvature. The Li-Yau result allows comparisons between dierent points at dierent times; the matrix result allows comparisons between dierent points at the same time also. However, their result is not entirelycontained in ours, in the sense that or general geometries the trace result can be obtained with restrictions only on the Ricci curvature. The matrix Harnack estimate given here is very useul or proving monotonicity ormulas, as in[gh]. We also include in this paper several estimates on positive solutions o the heat equation which are used in that paper, or which this one is intended to serve as a general reerence. 1. Let M be a compact maniold o dimension m and let / be a smooth solution o the heat equation on 0 < t < oo at with / > 0 or all time. We shall derive estimates on / that provide inormation on its behavior. The irst is a derivative bound. Theorem 1.1. Let be a positive solution with < A, and let K be a lower bound on the Ricci curvatures o M, so that Rij > Kgij with K > 0. Then t\dt<(l + 2Kt) 2 \og(a/). Proo. We compute J and Now i then a p/ a _ >/ 2 2 n n t DJDj OD DJDj dt A DiD / / } -2R. 13 /l0g(a//) = A(/l0gy) + dt J m ' ' \ * J _ t, ~l + 2Kt d (p + 2Kip < 1. at

4 116 RICHARD S. HAMILTON I we put then h > 0 at t = 0 and \D\ 2 h = <p l -p--log(a/) dh ^ A, as we see rom the previous computations, throwing away the square and bounding the Ricci curvature term with Rij > Kgij. It ollows rom the maximum principle that h < 0 or all time, and this is the theorem. We can use this result to get an upper bound on the solution / just in terms o its average value and the elapsed time. Corollary 1.2. Let be a positive solution to the heat equation. Then /(* *)< ^2 / M (x,t)dx or 0 < t < 1 with a constant B depending only on M. Proo. Consider the point (, T) in 0 < t < 1 where t m / 2 (x,t) assumes its maximum value, which we call 6. Then max t m/2 (x, t) = T m/2 /(, r) = b. On the restricted time interval r/2 < t < r we have /(#,i) < A where A = 2 m / 2 6/r m / 2. Applying Theorem 1.1 on this time interval ^\D(x,T)\ 2 < (l + ^T)/(x,r) 2 log(a/(x,t)) or all x <E M. Let and conclude that I we let e{x,t) = log(a/(x,t)) \DVI(X,T)\<^ 1 + KT 2T 1 + KT then or all x in the ball d(x, ) < p we have Ve(x,T)<V (z,t) + i/v2.

5 A MATRIX HARNACK ESTIMATE 117 Also we have which shows that 777 t(z,t)=\og(am,t)) = j log 2 i(x,t)<c m or all x in the ball <i(a;, ) < p, where C m denotes a constant depending only on m. Consequently (x,t)>c m A where C m > 0 is another constant depending only on m. We integrate this estimate over the ball d(x, ) < p. Since p «V^" or small r, the ball will have a volume V > CMT ' 2 or some constant CM > 0 depending only on M as long as r < 1. This gives CMT m/2 A< [ {x,t)dx. JM Since the integral o / over M is conserved, this proves the corollary. We can now use this bound in the original estimate. Corollary 1.3. There exists constants B and C depending only on M, such that i is any positive solution o the heat equation with. = 1, then or 0<t< 1 t\d\ 2 <C 2 log(b/t / 2 ). Proo To estimate \D\ 2 at any time r, we apply Theorem 1.1 on r/2 <t<r where we use the bound rom Corollary 1.2. Next we prove a Harnack estimate similar to that in the paper o Li and Yau[LY]. We need the case o negative curvature. Theorem 2.1. Let be a positive solution o the heat equation on a compact maniold M whose Ricci curvature is bounded below by i?^ > Kgij ork > 0. Then ^l e -2Kt\Dl 2Ktrn e dt +e 2t / -

6 118 RICHARD S. HAMILTON Proo. We use d_\dl dt and estimate rom the trace. When this gives = A m* DiDj- DJDj' 2 DiDj- m* h= d J--e-^ dt DJDj >±(?l_ m \dt dh ^ A t 2 2Kt d \D\ 2 \ 2 at m \dt m 2t J -21L DjDj ' e 2X1 m 22 / where we use the term rom the time derivative alling on e~ 2Kt to cancel the term with the Ricci curvature, and throw away the term rom the time derivative alling on e 2Kt. Now when h < 0 we have 0<e 2Kt m < F -2Kt\Di o \D\* d 2t J - dt - dt and hence dh - > Ah when h<0. at But h > +oo as t 0, so h > 0 or all time by the maximum principle. Corollary 2.2. /// is a positive solution to the heat equation or t > 0 and K is a lower bound or the Ricci curvature then or any points and X in M and any times r and T with 0 < r < T we have m,r)< (^y /2 (X,T) exp {ie^[l + 2K(T - r))^l + ynr _ ^ j. Proo. Let i = log /. The previous theorem says m -2Kt m dt

7 A MATRIX HARNACK ESTIMATE 119 Choose a path x = x(t) with x = at t r and x = X at t = T. Then integrating along the path and i we use the previous estimate and we get e- 2Kt \Dl\ 2 JlKt + Dl^->dt - 4 1{X,T)-1&T)>-\ J" J2Kt dx dt dx dt dt -Jr 2t l e ZKt dt. Now we choose the path rom to X to lie along a geodesic, but with arc length 8 = a [e- 2KT - e- 2Kt ] where a is a constant. In order to start at with s = 0 at t = r and end at X with 5 = d(x, ) at t = T we need d(x^) = a[e- 2KT -e- 2Kt ]. This makes n2kt dx 2K dt = dt -2KT _ p-2kt I' To make this look better, we estimate which makes and e u > 1 + u 1 < 1 + n 1-e- w 2K 2Ke 2KT e-2kt _ e -2KT I _ e -2K(T-T) Finally since we can estimate e u 1 < - + e" it u d(x,0 2. e T [l + 2K{T - r)] T-T T e2kt dt<log[^) + (e 2KT -e 2KT )- /

8 120 RICHARD S. HAMILTON Thus l&r) < e(x,t) + \e 2KT [l + 2K(T-T)]^ + log(^)+(e 2^-e 2^). I we exponentiate this we get the corollary. We can use this estimate to get a good lower bound on the undamental solution o the heat equation. Corollary 2.3. The undamental solution o the heat equation c(x,t, y) satisies k(x,t,y) > ^ exp-{1(1 + 2^)^ + e 2 -} or some constant c depending only on M. Proo It is well known that k(x,t,x)>c/t m/2. We apply this in the previous estimate taking {x,t) = k(x,tjy) and putting = y and letting r 0. D 3. We are now in a position to establish a basic comparison result that we need or the e-regularity result. Theorem 3.1. For any 6 > 0 and any constant C > 1 and or any X G M and any T with 0 < T < 1, there exists a p > 0 such that i 6 M satisies d{x, ) < p and r satisies T p 2 < r < T then or any positive solution {x,t) o the heat equation with = 1 T(Z,T)<6 + CT(X t T). Proo First note by Corollary 2.2 that /&T) < (^^/(^^exp {^ [e 2^ - e 2^]} and hence i p is small enough m,r)<c^,t)

9 A MATRIX HARNACK ESTIMATE 121 or any C > 1 we desire. Thus it suices to prove the estimate when T = r. Thus we must show mt)<t + C(X,T). For this we use the gradient estimate in Corollary 1.3, with i = T, so that T\D(x,T)\ 2 < K(x,T)nog(B/T m / 2 (x,t)) or all x. Let e = \og(b/t m / 2 (x,t)) and observe T\D \ 2 < KC and hence This makes I^I sw- I /(, T) < 6T, we are done, otherwise e(z,t)<iog B ST 2?- 1 Squaring the previous inequality gives t(x,t)<e(z,t) + pmlog(-, B \ 1 2 K ^1712-1J + 7P 4^ IF J 1 By choosing p small enough, we can make term 2 + term 3 < 77 or any 77 > 0. For such a choice o p we have e(x t T)<e(Z t T) + r, =* /(, r) < ev(x, T). < I + C/(X, T) with e 77 = C > 1. This proves the theorem.

10 122 RICHARD S. HAMILTON 4. Finally we come to the second derivative estimate. First we need a bound on A/. Lemma 4.1. There exists a constant C depending only on M such that i is a positive solution o the heat equation with 0 < / < A then or 0 < t < 1 Proo. Again we use ta<c[l + log(a/)]. d p/l 2 dt - / m A/- \D\2 } + 2K \Dl where K is a lower bound on the Ricci curvature. This time we let (p = [e Kt - l]/ke Kt and let h = cp [A/ + D/ 7] - /[m + 4 log(a/)] and compute dt - m[ J using <p' + Kip = 1. Now we claim that dh -7-7 < Ah whenever h > 0 at and we see this by examining three cases. \Dl (i) I A/ < we are done, since <// > ' A.\ l}. 2 \ Il J (ii) I J-HP- < A/ < 3^ we are also done, since ip' < 1. \D\ 2 (iii) I 3 ^-^ < A/, then when h > 0 / A/ \D\> and since <// < 1 we are done completely. >A+m >^L V Since h < 0 at = 0, the maximum principle implies h < 0 or all t. Since e u 1 > u, it ollows rom the deinition o i that t < e Kt (p(t) < e K (p(t) or 0 < t < 1, and the result ollows.

11 A MATRIX HARNACK ESTIMATE 123 Corollary 4.2. There exist constants B and C depending only on M such that i is a positive solution to the heat equation with J < 1 then or 0<t<l ta<c[l + \og(b/t m ' 2 )}. Proo. We know that or some constant B we have t m^2 < B by Corollary 1.2. Apply Lemma 4.1 on the time interval r/2 < t < r using the bound j <; 2 m / 2 B/r m^2 = A and we get the desired conclusion. Theorem 4.3. Let M be a compact maniold. There exist constants B and C depending only on M such that i is a positive solution to the heat equation with < 1 then or 0 < t < 1 we have Proo Let D.Dj - ^^l + 1/^ + C[l + log^/"/ 2 /)]^- > 0 i,, = DiDj - ^j^-. Then in a straightorward manner we compute d 2 Hij = AHij + -r-hij + 2Rikji H k i Ri k Hj k Rj k Hi k + - RikjiDkDi + [DtRtj - DiRl - DjRu] DJ where Hj = Hi k Hj k is the square o the matrix. Let = m + log(b/t m/2 /) where B is a constant with t m / 2 / < B rom Corollary 1.2, and let P=^t + A(K + L 2 / 3 ) where K is a lower bound or the sectional curvature o M, L is an upper bound or \DiRj k \ and A is a constant to be determined later. We compute where Q = -^- ^ + L^) + A(K + L^)\^l

12 124 RICHARD S. HAMILTON Note that jp 2 > 2^/ + ±(K + L^)i + 2A\K + L^je and since t > m we get Now let P 2 + Q > ^(^ + L2/3 )/^ + 2A 2 m(k + L 2/Z t. J t Ny = Hij + Pgtj and compute iv y = ANtj + jn* - IpNtj + 2RikjiNki RikNjk RjkNik + jrikiid k Pi + [DeRij - DiRjt - DjR^DJ + 7 P * + Q 9ij I the sectional curvatures o M are bounded below by K, then Pikji = Rikji + Kigijdki 9ii9jk) is positive in the sense that PikjtViWkVjWt > 0 or all vectors v and w. This makes RikjeNkt > PikjeNke - KNgy + KNy where N is the trace o iv^. Also we have RikjiDkDe > -KlDgij and i \DiRjk\ < L then [DeRij - DiRji - DjRu] Dr > -3L >/ ^. Also we can bound A < 2/3^p + L 4/3 /-

13 A MATRIX HARNACK ESTIMATE 125 Assembling these estimates, we get %-Nij > AN- + ^Nl - ^PNu + 2P ikjl N kl + 2KN ij - R ik N jk - R jk N ik + Zgij where Z=lpi + Q- 2KN- (2K + 3L 2 / 3 )^^ - 3L 4/3 /. Now observe that y + A{K + L^)i By Corollary 4.2 we have a bound or some constant C (and the same B). We also have / - t' or some constant C rom Corollary 1.3. Remembering > m, this makes 2KN + (2K + 3L 2/3 )^L + 3L 4/3 < -{K + l}' z )t + CA{K + L 2 ' z )e or some constant C depending only on M. Now i we choose A large enough with respect to C, we can make Z > 0. Then the maximum principle or matrices (see Hamilton[H]) implies that Nij > 0 or all time. We only have to check that iv^- > 0 at t = 0, and that each o the terms Nj, PNr P m N ku KNy, R ik N jk + R jk N ik is non-negative at a null-eigenvector o iv^. This is immediate except perhaps or Pi k jin k i, but here it is also clear i we diagonalize N k i with respect to a basis. This completes the proo. Corollary 4.4. Suppose M has non-negative sectional curvature and is Ricci parallel, so that Ri k ji > 0 and DiRj k = 0. Then or any positive solution o the heat equation _ DiDj, 1. ^n DiDj j-l- + gy > 0.

14 126 RICHARD S. HAMILTON Proo. The above proo simpliies, taking K = L = 0. This result applies on lat tori, on spheres, on complex projective spaces, on a product o such spaces, and on a quotient o a product. REFERENCES [GH] Grayson, M. and Hamilton, R. S., The ormation o singularities in the harmonic map heat low, preprint, UCSD. [H] Hamilton, R. S., Four-maniolds with positive curvature operator, J. Dierential Geom. 24 (1986), [LY] Li, P., and Yau, S.-T., On the parabolic kernel o the Schrodinger operator. Acta Math. 156 (1986), UNIVERSITY OF CALIFORNIA, SAN DIEGO, U. S. A. RECEIVED JULY 10, 1992