The Logarithmic Sobolev Inequality Along The Ricci Flow (revised version)

Transcription

1 arxiv: v [math.dg] Jul 007 The Logarithmic Sobolev Inequality Along The Ricci Flow (revised version) Rugang Ye Department of athematics University of California, Santa Barbara July 0, 007. Introduction. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow 6. The κ-noncollapsing estimate Appendix A. The logarithmic Sobolev inequalities on the euclidean space Appendix B. The estimate of e th Introduction Consider a compact manifold of dimension n 3. Let g = g(t) be a smooth solution of the Ricci flow g t = Ric (.) on [0, T) for some (finite or infinite) T > 0 with a given initial metric g(0) = g 0. Theorem A For each σ > 0 and each t [0, T) there holds u ln u dvol σ ( u + R 4 u )dvol n ln σ + A (t + σ 4 ) + A (.)

2 for all u W, () with u dvol =, where A = 4 C S (, g 0 ) vol g0 () n min R g0, A = n ln C S (, g 0 ) + n (ln n ), and all geometric quantities are associated with the metric g(t) (e.g. the volume form dvol and the scalar curvature R), except the scalar curvature R g0, the modified Sobolev constant C S (, g 0 ) (see Section for its definition) and the volume vol g0 () which are those of the initial metric g 0. Consequently, there holds for each t [0, T) u ln u dvol n [α ln I ( ( u + R 4 u )dvol + A ] 4 ) (.3) for all u W, () with u dvol =, where α I = e n e(at+a) n. (.4) The exact factor n in the term n ln σ in the logarithmic Sobolev inequality (.) (also in (.5) and (.8) below) is crucial for the purpose of Theorem D. Note that an upper bound for the Sobolev constant C S (, g 0 ) and the modified Sobolev constant C S (, g 0 ) can be obtained in terms of a lower bound for the diameter rescaled Ricci curvature and a positive lower bound for the diameter rescaled volume, see Section. In particular, a lower bound for the Ricci curvature, a positive lower bound for the volume and an upper bound for the diameter lead to an upper bound for the Sobolev constant and the modified Sobolev constant. The logarithmic Sobolev inequality in Theorem A is uniform for all time which lies below a given bound, but deteriorates as time becomes large. The next result takes care of large time under the assumption that a certain eigenvalue λ 0 of the initial metric is positive. This assumption holds true e.g. when the scalar curvature is nonnegative and not identically zero. Theorem B Assume that the first eigenvalue λ 0 = λ 0 (g 0 ) of the operator + R 4 for the initial metric g 0 is positive. Let δ 0 = δ 0 (g 0 ) be the number defined in (3.). Let t [0, T) and σ > 0 satisfy t + σ nc 8 S(, g 0 ) δ 0. Then there holds u ln u dvol σ ( u + R 4 u )dvol n ln σ + n ln n + n ln C S(, g 0 ) + σ 0 (g 0 ) (.5)

3 for all u W, () with u dvol =, where all geometric quantities are associated with the metric g(t) (e.g. the volume form dvol and the scalar curvature R), except the Sobolev constant C S (, g 0 ) and the number σ 0 (g 0 ) (defined in (3.3)) which are those of the initial metric g 0. Consequently, there holds for each t [0, T) u ln u dvol n [ ln α II ( u + R ] 4 u )dvol (.6) for all u W, () with u dvol =, where α II = ec S (, g 0 ) e n σ 0(g 0 ). (.7) Combining Theorem A and Theorem B we obtain a uniform logarithmic Sobolev inequality along the Ricci flow without any restriction on time or the factor σ. Theorem C Assume that λ 0 (g 0 ) > 0. For each t [0, T) and each σ > 0 there holds u ln u dvol σ ( u + R 4 u )dvol n ln σ + C (.8) for all u W, () with u dvol =, where C depends only on the dimension n, a positive lower bound for vol g0 (), a nonpositive lower bound for R g0, an upper bound for C S (, g 0 ), and a positive lower bound for λ 0 (g 0 ). Consequently, there holds for each t [0, T) u ln u dvol n [ ln α III ( u + R ] 4 u )dvol (.9) for all u W, () with u dvol =, where α III = e n e n C. (.0) We note here a special consequence of Theorem C. Corollary Assume that λ 0 (g 0 ) > 0. Then we have at any time t [0, T) vol g(t) () e 4 C (.) when ˆR(t) 0, and vol g(t) () e 4 C ˆR(t) n (.) 3

4 when ˆR(t) > 0. Similar volume bounds follow from Theorem A without the condition λ 0 (g 0 ) > 0, but they also depend on a (finite) upper bound of T. The class of Riemannian manifolds (, g 0 ) with λ 0 (g 0 ) > 0 is a very large one and particularly significant from a geometric point of view. On the other hand, we would like to point out that the assumption λ 0 (g 0 ) > 0 in Theorem C is indispensible in general. In other words, a uniform logarithmic Sobolev inequality like (.8) without the assumption λ 0 (g 0 ) > 0 is false in general. Indeed, by [HI] there are smooth solutions of the Ricci flow on torus bundles over the circle which exist for all time, have bounded curvature, and collapse as t. In view of the proofs of Theorem D and Theorem E, a uniform logarithmic Sobolev inequality like (.8) fails to hold along these solutions. The generalization of Theorem C stated in the first posted version of this paper is thus incorrect. The trouble with the proof, which we found in the process of trying to work out an improvement of the logarithmic Sobolev inequality along the Ricci flow, stems from the application of a monotonicity formula in [Z]. ore precisely, for a given T > 0, the estimate F e f (R + f ) (4πτ) n dvol n τ (.3) in [Z] (needed for proving the monotonicity of the generalized entropy in [Z]) holds true only for a solution f of the equation (4.8) defined up to T, where τ = T t. For the choice T = t 0 for a given t 0 in [Z] and the first posted version of this paper, where t 0 is denoted t, f is assumed to start backwards at t 0, and may not exist on [t 0, T ). Hence the inequality (.3) may not hold, and the generalized entropy may not be monotone. In contrast, Perelman s entropy monotonicity is always valid in any time interval where f satisfies (4.8). In other words, for a given t 0, one can choose T > t 0 arbitrarily to define τ = T t. Perelman s entropy is monotone on [0, t 0 ] as long as f satisfies (4.8) there. This is crucial for applying Perelman s entropy. For a brief account of the logarithmic Sobolev inequalitities on the euclidean space we refer to Appendix A, which serve as the background for the idea of the logarithmic Sobolev inequality. Both Theorem A and Theorem B are consequences of Perelman s entropy monotonicity [P]. We obtained these two results and Theorem C in 004 (around the time of the author s differential geometry seminar talk An introduction to the logarithmic Sobolev inequality at UCSB in June 004). They have also been prepared as part of the notes [Y3]. Inspired by an argument in [Z], we apply the results as presented in [D] to derive from Theorem D a Sobolev inequality along the Ricci flow without any restriction on time. A particularly nice feature of the results in [D] is that no additional geometric data (such as the volume) are involved in the passage from the logarithmic Sobolev inequality to the Sobolev inequality. Only the non-integral terms in the logarithmic 4

5 Sobolev inequality and a nonpositive lower bound for the potential function Ψ (see Theorem 5.3) come into play. This leads to the form of the geometric dependenc in the following theorem. Theorem D Assume that λ 0 (g 0 ) > 0. There is a positive constant A depending only on the dimension n, a nonpositive lower bound for R g0, a positive lower bound for vol g0 (), an upper bound for C S (, g 0 ), and a positive lower bound for λ 0 (g 0 ), such that for each t [0, T) and all u W, () there holds ( ) n u n n n dvol A ( u + R 4 u )dvol, (.4) where all geometric quantities except A are associated with g(t). In a similar fashion, a Sobolev inequality follows from Theorem A in which the condition λ 0 (g 0 ) > 0 is not assumed, but the bounds also depend on an upper bound of time. Theorem D Assume T <. There are positive constants A and B depending only on the dimension n, a nonpositive lower bound for R g0, a positive lower bound for vol g0 (), an upper bound for C S (, g 0 ), and an upper bound for T, such that for each t [0, T) and all u W, () there holds ( ) n u n n n dvol A ( u + R 4 u )dvol + B u dvol, (.5) where all geometric quantities except A and B are associated with g(t). Next we deduce from Theorem D a κ-noncollapsing estimate for the Ricci flow for all time which improves Perelman s κ-noncollapsing result [P] for bounded time. Our estimate is independent of time and hence is uniform for all time. In particular, it holds both in a finite time interval and an infinite time interval. oreover, our estimate provides a clear and uniform geometric dependence on the initial metric which appears to be optimal qualitatively. The κ-noncollapsing estimate below is measured relative to upper bounds of the scalar curvature. The original κ-noncollapsing result of Perelman in [P] is formulated relative to bounds for Rm. Later, a κ-noncollapsing result for bounded time measured relative to upper bounds of the scalar curvature was obtained independently by Perelman (see [KL]) and the present author (see [Y]). Theorem E Assume that λ 0 (g 0 ) > 0. Let t [0, T). Consider the Riemannian manifold (, g) with g = g(t). Assume R r on a geodesic ball B(x, r) with r > 0. 5

6 Then there holds vol(b(x, r)) ( ) n r n, (.6) n+3 A where A is from Theorem D. In other words, the flow g = g(t), t [0, T) is κ- noncollapsed relative to upper bounds of the scalar curvature on all scales. A similar κ-noncollapsing estimate for bounded time follows from Theorem D, for which the condition λ 0 (g 0 ) > 0 is not assumed. Theorem E Assume that T <. Let L > 0 and t [0, T). Consider the Riemannian manifold (, g) with g = g(t). Assume R on a geodesic ball B(x, r) with r 0 < r L. Then there holds ( ) n vol(b(x, r)) r n, (.7) n+3 A + BL where A and B are from Theorem D. This theorem improves the previous κ-noncollapsing results for bounded time mentioned above. Namely it provides an explicit estimate with clear geometric dependence. oreover, the estimate is uniform up to t = 0 (under a given upper bound for T). (Of course, this is also the case for Theorem E.) One special consequence of Theorem E is that one can obtain smooth blow-up limits at T both in the case T < and T =, assuming that g becomes singular at T. Previously, this was possible only at T < thanks to Perelman s κ-noncollapsing result. We formulate a theorem. Let a k be a sequence of positive numbers such that a k, and T k (0, T) with T k T. Consider the rescaled Ricci flows g k (t) = a k g(t k + a k t) on ( a kt k, 0]. Theorem F Assume that λ 0 (g 0 ) > 0. Assume that a k T k T < 0. oreover, assume that there is a sequence of points x k with the following property. For each L > 0 there is a positive constant K such that Rm K holds true for g k (t) on the geodesic ball of center x k and radius K, where t ( a k T k, 0] is arbitrary. Then a subsequence of ( ( a k T k, 0], g k, x k ) point converges smoothly to a pointed Ricci flow ( ( T, 0], g, x ) for some manifold and x, such that g (t) is complete for each t. The flow g is κ-noncollapsed relative to upper bounds of the scalar curvature on all scales, where κ = n(n+3) A n and A is from Theorem E. oreover, there holds for g at all t ( ) n u n n n dvol A ( u + R 4 u )dvol (.8) 6

7 for all u W, (). By scaling invariance, the above results extend straightforwardly to the modified Ricci flow g t = Ric + λ(g, t)g (.9) with a smooth scalar function λ(g, t) independent of x. The volume-normalized Ricci flow g t = Ric + n ˆRg (.0) on a closed manifold, with ˆR denoting the average scalar curvature, is an example of the modified Ricci flow. The λ-normalized Ricci flow g t = Ric + λg (.) for a constant λ is another example. (Of course, it reduces to the Ricci flow when λ =.) The normalized Kähler-Ricci flow is a special case of it. We have e.g. the following result. Theorem G Theorem D and Theorem E extend to the modified Ricci flow. Theorem F also extends to the case of the volume-normalized Ricci flow and the λ-normalized Ricci flow, when the limit flow equation is defined accordingly. Theorem F actually extends to the general case of the normalized Ricci flow under an additional assumption on the convergence of the rescaled λ(g, t). We omit the statements. By Perelman s scalar curvature bound [ST] we obtain the following corollary. Theorem H Let g = g(t) be a smooth solution of the normalized Kähler-Ricci flow g t = Ric + γg (.) on [0, ) with a positive first Chern class, where γ is the positive constant such that the Ricci class equals γ times the Kähler class. (We assume that carries such a Kähler structure.) Assume that λ 0 (g 0 ) > 0, where g 0 = g(0). Then the Sobolev inequality (.5) holds true for g. oreover, there are positive constants L and δ depending only on the initial metric g 0 and the dimension n such that vol(b(x, r)) δr n (.3) 7

8 for all t [0, ), x and 0 < r L. Consequently, blow-up limits of g at the time infinity satisfy (.3) for all r > 0 and the Sobolev inequality ( ) n u n n n dvol A u dvol (.4) for all u, where A is from Theorem D. In particular, they must be noncompact. The results in this paper extend to dimension n =. This will be presented elsewhere. We would like to acknowledge that Guofang Wei first brought our attention to Zhang s paper [Z]. The Sobolev inequality Consider a compact Riemannian manifold (, g) of dimension n 3. Its Poincaré- Sobolev constant (for the exponent ) is defined to be C P,S (, g) = sup{ u u n n : u C (), u = }, (.) where u p denotes the L p norm of u with respect to g, i.e. u p = ( u p dvol) /p (dvol = dvol g ). In other words, C P,S (, g) is the smallest number such that the Poincare-Sobolev inequality u u n n C P,S (, g) u (.) holds true for all u C () (or all u W, ()). The Sobolev constant of (, g) (for the exponent ) is defined to be C S (, g) = sup{ u n n u vol() : u C (), u =. (.3) n In other words, C S (, g) is the smallest number such that the inequality u n n holds true for all u W, (). C S (, g) u + u vol() (.4) n Definition We define the modified Sobolev constant C S (, g) to be max{c S (, g), }. The Hölder inequality leads to the following basic fact. 8

9 Lemma. There holds for all u W, () u n n C P,S (, g) u + u vol(). (.5) n In other words, there holds C S (, g) C P,S (, g). Another basic constant, the Neumann isoperimetric constant of (, g), is defined to be C N,I (, g) = sup{ n vol(ω) n A( Ω) : Ω is a C domain, vol(ω) vol()}, (.6) where A( Ω) denotes the n -dimensional volume of Ω. Lemma. There holds for all u W, () u u n n 4(n ) n C N,I(, g) u. (.7) In other words, there holds C P,S (, g) 4(n ) n C N,I(, g). For the proof see [Y]. The following estimate of the Neumann isoperimetric constant follows from S. Gallot s estimate in [Ga]. We define the diamater rescaled Ricci curvature ˆRic(v, v) of a unit tangent vector v to be diam() Ric(v, v), and set κ ˆRic = min v{ ˆRic(v, v)}. Then we set ˆκ ˆRic = min{κ ˆRic, }. We also define the diameter rescaled volume ˆvol() to be vol()diam() n. Theorem.3 There holds C N,I (g, ) C(n, ˆκ ˆRic )ˆvol() n, (.8) where C(n, ˆκ ˆRic ) is a positive constant depending only on n and ˆκ ˆRic. Note that ˆκ ˆRic can be replaced by a certain integral lower bound of the Ricci curvature, see [Ga]. 3 The Logarithmic Sobolev inequalities on a Riemannian anifold The various versions of the logarithmic Sobolev inequality on the Euclidean space as presented in Appendix A allow suitable extentions to Riemannian manifolds. We formulate a log gradient version and a straight version, cf. Appendix A. As in the last section, let (, g) be a compact Riemannian manifold of dimension n. 9

10 Theorem 3. There holds ( ) u ln u dvol n ln C S (, g) u +, (3.) vol g () n provided that u W, () and u =. Proof. Set q = n. Since ln is concave and n u dvol =, we have by Jensen s inequality ln u q dvol = ln u u q dvol u ln u q. (3.) It follows that u ln u = q ln u q dvol q q ln u q ) (C n ln S (, g) u + u vol g (). (3.3) n Lemma 3. There holds for all B 0, α > 0 and x > B. ln(x + B) αx + αb lnα (3.4) Proof. Consider the function y = ln(x + B) αx for x > B. Since y as x B or x, it achieves its maximum somewhere. We have y = α. (3.5) x + B Hence the maximum point is x 0 = α y(x 0 ) = αb ln α. B. It follows that the maximum of y is Theorem 3.3 For each α > 0 and all u W, () with u = there holds u ln u nαc S(, g) u n ln α + n (ln + αvol g() n ) (3.6) 0

11 and u ln u nαc S(, g) + nα (vol g() n min R (The notation of the volume is omitted.) ( u + R 4 u ) n lnα Proof. By (3.) we have for u W, () with u = 4 C S (, g) ) + n (ln ). (3.7) u ln u (C n ln S (, g) u + vol g () n ( n ln + n ) ln C S (, g) u +. (3.8) vol g () n Applying Lemma 3. with x = C S (, g) u and B = we then arrive at (3.6). The inequality (3.7) follows from (3.6). ) Lemma 3.4 Let A > 0, B > 0 and γ > 0 such that A. Then we have γ+b for all x γ. ln(x + B) Ax ln A + ln(γ + B) ln γ (3.9) Proof. First consider the function y = lnt γt for t > 0. Since y as t 0 or t, y achieves its maxmum somewhere. We have y = γ. Hence the maximum t is achieved at. It follows that the maximum is y( ) = ln γ. We infer γ γ ln A γa ln γ. (3.0) Next we consider the function y = ln(x + B) Ax + ln A for x γ. By (3.0) we have y(γ) = ln(γ + B) Aγ + ln A ln(γ + B) ln γ. On the other hand, we have y = A A 0. We arrive at (3.9). x+b γ+b Theorem 3.5 Assume that the first eigenvalue λ 0 = λ 0 (g) of the operator + R 4 is positive. For each A δ 0 and all u W, () with u = there holds u ln u nac S ( u + R 4 u ) n ln A + n ln + σ 0, (3.)

12 where σ 0 = σ 0 (g) == n and C S = C S (, g). δ 0 = δ 0 (g) = (λ 0 C S + vol g () n [ ln(λ 0 C S + vol g () n CS min R ), (3.) 4 ] CS min R ) ln(λ 0 CS 4 ), (3.3) Proof. Arguing as in the proof of Theorem 3.3 we deduce for u W, () with u = u ln u n ln + n ln(c S u + ) vol g () n n ln + n ] [C ln S ( u + R 4 u ) + C min R vol g () S. (3.4) n 4 Applying (3.9) with γ = λ 0 CS, B = C vol g() n S minr and x = C 4 S ( u + R 4 u ) we then arrive at (3.) for each A (γ + B). 4 The logarithmic Sobolev inequality along the Ricci flow Let be a compact manifold of dimension n. Consider Perelman s entropy functional [ W(g, f, τ) = τ(r + f ) + f n ] e f dvol, (4.) (4πτ) n where τ is a positive number, g is a Riemannian metric on, and f C () satisfies e f dvol =. (4.) (4πτ) n All goemetric quantities in (4.) and (4.) are associated with g. To relate to the idea of logarithmic Sobolev inequalities we make a change of variable u = f e. (4.3) (4πτ) n 4

13 Then (4.) leads to and we have u dvol = (4.4) where W(g, f, τ) = W (g, u, τ) n ln τ n ln(4π) n (4.5) W (g, u, τ) = [ τ(4 u + Ru ) u ln u ] dvol. (4.6) We define µ (g, τ) to be the infimum of W (g, u, τ) over all u satisfying (4.4). Next let g = g(t) be a smooth solution of the Ricci flow g t = Ric (4.7) on [0, T) for some (finite or infinite) T > 0. Let 0 < t < T and σ > 0. We set T = t + σ and τ = τ(t) = T t for 0 t t. Consider a solution f = f(t) of the equation f t = f + f R + n τ (4.8) on [0, t ] with a given terminal value at t = t (i.e. τ = σ) satisfying (4.) with g = g(t ). Then (4.) holds true for f = f(t), g = g(t) and all t [0, t ]. Perelman s monotonicity formula says dw dt = τ where W = W(g(t), f(t), τ(t)). Consequently, where g = g(t), τ = τ(t) and which satisfies the equation Ric + f e f τ g dvol 0, (4.9) (4πτ) n d dt W (g, u, τ) n d ln τ, (4.0) dt u = u(t) = e f(t)/, (4.) (4πτ(t)) n 4 u t = u + u u + R u. (4.) 3

14 It follows that µ (g(t ), τ(t )) µ (g(t, τ(t )) + n ln τ τ, (4.3) for t < t, where τ = τ(t ) and τ = τ(t ). Choosing t = 0 and t = t we then arrive at µ (g(0), t + σ) µ (g(t ), σ) + n ln t + σ σ. (4.4) Since 0 < t < T is arbitrary, we can rewrite (4.4) as follows µ (g(t), σ) µ (g(0), t + σ) + n ln σ t + σ for all t [0, T) and σ > 0 (the case t = 0 is trivial). We ll also need the following elementary lemma. (4.5) Lemma 4. Let a > 0 and b be constants. Then the minimum of the function y = aσ n ln σ + b for σ > 0 is n ln(αa), where α = e n eb n. (4.6) Proof. Since y as t 0 or t, it achieves its minimum somewhere. We have y = a n, whence the minimum is achieved at σ = n. Then the minimum σ a equals y( n ), which leads to the desired conclusion. a Proof of Theorem A We apply Theorem 3.3 with g = g 0 to estimate µ (g 0, t+σ). Consider u W, () with u =. We choose in (3.6) and deduce u ln u 4(t + σ) (t + σ) α = 8(t + σ) n C S (, g 0 ) (4.7) u n 8(t + σ) ln n C + n S 8(t + σ) n C S vol g 0 () n (4 u + Ru 4 ) + (t + σ)( n C S vol g 0 () n + n (ln ) min t=0 R) n ln(t + σ) + n ( ln C S + lnn ln ), (4.8) 4

15 where C S = C S (, g 0 ). It follows that µ (g(0), t + σ) n ln(t + σ) (t + σ)( 4 n C S vol g 0 () n Combining this with (4.5) leads to min t=0 R) n ( ln C S + ln n ln ). (4.9) µ (g(t), σ) n ln σ (t + σ)( 4 n C S vol g() n min t=0 R) n ( ln C S + ln n ln ), (4.0) or µ (g(t), σ 4 ) n ln σ (t + σ 4 )( 4 n C S vol g() n min t=0 R) n ( ln C S + ln n ), (4.) which is equivalent to (.). To see (.3) we apply Lemma 4. to (.) with a = ( u + R 4 u )dvol+ A 4 and b = A t + A. Proof of Theorem B This is similar to the proof of Theorem A. We apply Theorem 3.5 with g = g 0 to estimate µ (g 0, t + σ). Assume t + σ n 8 C S(, g 0 ) δ 0 (g 0 ). We set A = 8(t + σ) nc S (, g 0 ). (4.) Then there holds A δ 0 (g 0 ). Using this A in (3.) we deduce for u W, () with u = u ln u 4(t + σ) ( u + R 4 u ) n ln(t + σ) It follows that + n ( lnc S(, g 0 ) + ln n ln) + σ 0 (g 0 ). (4.3) µ (g 0, t + σ) n ln(t + σ) n ( ln C S(, g 0 ) + ln n ln) σ 0 (g 0 ). (4.4) Combining this with (4.5) yields µ (g(t), σ) n ln σ n ( lnc S(, g 0 ) + ln n ln) σ 0 (g 0 ). (4.5) 5

16 Replacing σ by σ we then arrive at (.5). 4 To see (.6), we apply Lemma 4. to (.5) with a = ( u + R 4 u ) and b = n ln n + n ln C S(, g 0 ) + σ 0 (g 0 ). Proof of Theorem C Consider t [0, T) and σ > 0. If σ < n 8 C S(, g 0 ) δ 0 (g 0 ), we apply Theorem A. Otherwise, we apply Theorem B. Then we arrive at (.8). To see (.9), we apply Lemma 4. to (.8) with a = ( u + R 4 u ) and b = C. Proof of Corollary to Theorem C Choosing u = vol g(t) () in (.8) we infer ln vol g(t) () σ 4 ˆR(t) n ln σ + C. (4.6) If ˆR(t) 0 we choose σ = to arrive at (.). If ˆR(t) > 0, we choose σ = ˆR(t) to arrive at (.). 5 The Sobolev inequality along the Ricci flow We first present a general result which converts a logarithmic Sobolev inequality to a Sobolev inequality. It follows straightforwardly from more general results in [D]. Consider a compact Riemannian manifold (, g) of dimension n. Let Ψ L (), which we call a potential function. We set H = + Ψ. Its associated quadratic form is Q(u) = ( u + Ψu )dvol, (5.) where u W, (). We also use Q to denote the corresponding bilinear form, i.e. Q(u, v) = ( u v + Ψuv)dvol. (5.) Consider the operator e th associated with H. There holds e th u 0 = K(, y, t)u 0 (t)dvol, (5.3) where K(x, y, t) denotes the heat kernel of H. We also have the spectral formula e th = e λ it φ i < φ i, >, where {φ i } is a complete set of L -orthonormal eigenfunctions of H and λ λ are the corresponding eigenvalues. 6

17 Theorem 5. Let σ > 0. Assume that for each 0 < σ σ the logarithmic Sobolev inequality u ln u dvol σq(u) + β(σ) (5.4) holds true for all u W, () with u =, where β is a non-increasing continuous function. Assume that τ(t) = t t is finite for all 0 < t σ. Then there holds 0 β(σ)dσ (5.5) e th u e τ(t) 3t 4 minψ u (5.6) for each 0 < t 4 σ and all u L (). There also holds for each 0 < t 4 σ and all u L (). e th u e τ( t ) 3t 4 minψ u (5.7) Note that (5.7) is equivalent to an upper bound for the heat kernel. This theorem follows from [D, Corollary..8] or rather its proof in [D]. The nonincreasing condition on β can easily be removed (the function τ(t) needs to be slightly modified). [D, Corollary..8] is formulated for the global case in which (5.4) is assumed for all σ > 0, but its proof in [D] actually covers the local case 0 < σ σ. For the convenience of the reader, we reproduce this proof in Appendix B. The global case is customarily phrased in terms of ultracontractivity, i.e. the logarithmic Sobolev inequality implies the ultracontractivity of e th, see e.g. [D]. Note that the global case suffices for the main purpose of this paper. Theorem 5. ) Let µ >. Then the inequality e th u c t µ 4 e (min Ψ )t u (5.8) holds true for each 0 < t < and all u L () if and only if the Sobolev inequality u µ µ c (Q(u) + ( min Ψ ) u ) (5.9) holds true for all u W, (), where c = c (c, µ ) and c = c (c, µ ). ) Assume Ψ 0 and let µ >. Then the inequality e th u c t µ 4 u (5.0) holds true for each t > 0 and all u L () if and only if the Sobolev inequality u µ µ c Q(u) (5.) holds true for all u W, (), where c = c (c, µ ) and c = c (c, µ ). 7

18 Proof. Part ) is contained in [D, Theorem.4.], which is based on the Riesz-Thorin interpolation theorem and the arcinkiewicz interpolation theorem. Part ) in the case Ψ 0 is contained in [D, Corollary.4.3]. It follows from Part ) since the operator H + satisfies the condition of Part ) (one reduces a large time to the time interval (0, ) by iterations over intervals of length and ). The general case of Part ) follows from this special case by considering the operator H min Ψ. Combining Theorem 5. and 5. we arrive at the following result. Theorem 5.3 Let σ > 0. Assume that for each 0 < σ < σ the logarithmic Sobolev inequality u lnu dvol σq(u) µ ln σ + C (5.) holds true for all u W, () with u =, where µ and C are constants such that µ >. Then we have the Sobolev inequality u µ µ ( σ 4 ) n µ c (Q(u) + 4 σ min Ψ σ u for all u W, (), where c = c( C, µ ) and C is defined in (5.7) below. ) (5.3) Proof. For λ > 0 we consider the metric ḡ = λ g and the potential function Ψ = λ Ψ. Let H = ḡ + Ψ and Q the associated quadratic form. It follows from (5.) that u lnu dvolḡ σ Q(u) µ ln σ + (n µ) ln λ + C (5.4) for 0 < σ < λ σ and u W, () with u =. Choosing λ = σ we obtain u ln u dvolḡ σ Q(u) µ ln σ + n µ (ln σ ln) + C (5.5) for each 0 < σ < 4. By Theorem 5. we have for each 0 < t < and u L () where e th u Ct µ 4 u,ḡ, (5.6) C = µ n (σ ) n µ 4 e µ 4 3σ 6 minψ + C. (5.7) Applying Theorem 5. and converting back to g we then arrive at (5.3). 8

19 Proof of Theorem D Applying Theorem C and Theorem 5.3 with Ψ = R,µ = n and 4 σ = 4 we deduce ( u n c ( u + R n 4 u )dvol + ( min t R ) ) u dvol, (5.8) 4 where c = c(c, min t R ). By the maximum principle, we have min t R min t=0 R. Hence we arrive at ( u n c ( u + R n 4 u )dvol + ( min 0 R ) ) u dvol (5.9) 4 with c = c(c, min 0 R ). Since λ 0 is nondecreasing along the Ricci flow [P], we obtain u n c( + n λ 0 (g 0 ) ( min ( 0 R )) ( u + R ) 4 4 u )dvol (5.0) which leads to (.5). Proof of Theorem D This is similar to the above proof. 6 The κ-noncollapsing estimate It is obvious that Theorem E and Theorem E follow from Theorem D, Theorem D and the following lemma. Lemma 6. Consider the Riemannian manifold (, g) for a given metric g, such that for some A > 0 and B > 0 the Sobolev inequality ( ) n u n n n dvol A ( u + R 4 u )dvol + B u dvol (6.) holds true for all u W, (). Let L > 0. Assume R on a geodesic ball B(x, r) r with 0 < r L. Then there holds ( vol(b(x, r)) n+3 A + BL ) n r n. (6.) Proof. Let L > 0. Assume that R r on a closed geodesic ball B(x 0, r) with 0 < r L, but the estimate (6.) does not hold, i.e. vol(b(x 0, r)) < δr n, (6.3) 9

20 where ( δ = n+3 A + BL ) n. (6.4) We derive a contradiction. Set ḡ = r g. Then we have for ḡ vol(b(x 0, )) < δ (6.5) and R on B(x 0, ). oreover, (6.) leads to the following Sobolev inequality for ḡ ( ) n u n n n A ( u + R 4 u ) + BL u, (6.6) where the notation of the volume form is omitted. For u C () with support contained in B(x 0, ) we then have ( ) n u n n n B(x 0,) By Hölder s inequality and (6.5) we have B(x 0,) A ( u + B(x 0,) 4 u ) + BL u. (6.7) B(x 0,) u δ n ( u n n B(x 0,) ) n n. (6.8) Hence we deduce ( ) n u n n n B(x 0,) A u + ( A ( ) n B(x 0,) 4 + BL )δ n u n n n B(x 0,) ( ) n A B(x 0,) u + u n n B(x 0,) n. (6.9) It follows that ( ) n u n n n B(x 0,) A u. (6.0) B(x 0,) Next consider an arbitrary domain Ω B(x 0, ). For u C (Ω) with support contained in Ω we deduce from (6.0) via Hölder s inequality u Avol(Ω) n u. (6.) B(x 0,) Ω 0

21 Hence we arrive at the following Faber-Krahn inequality λ (Ω)vol(Ω) n A, (6.) where λ (Ω) denotes the first Dirichlet eigenvalue of on Ω. By the proof of [C, Proposition.4] in [C] we then infer vol(b(x, ρ)) ( ) n ρ n n+3 A (6.3) for all B(x, ρ) B(x 0, ). Consequently we have vol(b(x 0, )) ( ) n, (6.4) n+3 A contradicting (6.5). For the convenience of the reader, we reproduce here the arguments in the proof of [C, Proposition.4] in [C]. Consider B(x, ρ) B(x 0, ). Set u(y) = ρ d(x, y). Then we obtain λ (B(x, ρ)) λ (int B(x, ρ)) By (6.) we then infer Iterating (6.6) we obtain vol(b(x, r)) B(x,ρ/) u 4vol(B(x, ρ)) ρ vol(b(x, ρ/)). (6.5) ( ) n ρ n+ vol(b(x, ρ)) 4 n ρ n+ vol(b(x, A )) n n+. (6.6) ( ) P m ρ l= ( n n+ )l vol(b(x, ρ)) 4 P m l= l( n ρ n+ )l n vol(b(x, A m))( n+ )m (6.7) for all natural numbers m. Letting m we finally arrive at vol(b(x, ρ)) = ( ρ A ( ρ A ) P l= ( n n+ )l 4 P l= l( n n+ )l ) n ( ) n 4 n(n+) 4 = ρ n. (6.8) n+3 A Proof of Theorem F This theorem follows from Theorem E and Cheeger-Gromov- Hamilton compactness theorem.

22 Appendices A The logarithmic Sobolev inequalities on the Euclidean space In this appendix we review several versions of the logarithmic Sobolev inequality on the euclidean space for the purpose of presenting the background of the logarithmic Sobolev inequalitites. These versions are equivalent to each other.. The Gaussian version This is the original version of L. Gross. Theorem A. Let u W, loc (Rn ) satisfy R n u dµ =, where dµ = (π) n e x dx. (A.) Then u ln u dµ u dµ. R n R n (A.). The straight (Euclidean volume element) version Theorem A. There holds u ln u dx u dx, (A.3) provided that u W, (R n ) and u dx = (π) n/ e n. Equivalently, for β > 0, u ln u dx u dx + β ln β n β ln(πe ), (A.4) provided that u W, (R n ) and u = β. 3. The log gradient version It appears to be stronger than the other versions because of the logarithm in front of the Dirichlet integral of u.

23 Theorem A.3 There holds u ln u dx n [ ln πne ] u dx, (A.5) provided that u W, (R n ) and u dx =. 4. The entropy version (as formulated in [P]) This version is intimately related to Perelman s entropy functional W. Indeed, it can be viewed as the motivation for W. Theorem A.4 There holds provided that f W, loc (Rn ) and e f dx = (π) n/. B The estimate for e th ( f + f n)e f dx 0, (A.6) In this appendix we present, for the convenience of the reader, the detailed proof of Theorem 5.. It is basically a reproduction of the correspoding proof in [D]. Using the spectral formula for e th one readily shows that e th is a bounded operator on L () for all t 0. Indeed, using this formula and basic elliptic estimates one also shows that e th is a bounded operator on L p () for each p and all t 0. The maximum principle implies that e th is positivity preserving and a contraction on L () for all t 0, and hence e th is a symmetric arkov semigroup on L () (see [D, p.] for the definition), provided that Ψ 0. Consequently, the general results in Chapter of [D] can be applied. (The proof below is self-contained and does not appeal to this concept.) By duality and interpolation, e th is a contraction on L p () for all p and t 0, if Ψ 0 (see [D, Theorem.3.3]). Using the spectral formula for e th one also sees that e th is a contraction on W, (), if Ψ 0. Proof of Theorem 5. Part We first assume Ψ 0, i.e. min Ψ = 0. It follows from (5.4) u ln u σq(u) + β(σ) u + u ln u (B.) 3

24 for all u W, (). Here the notation of the volume form is omitted. Replacing u by u p/ for p > and u W, () L () we deduce p u p ln u σq( u p ) + β(σ) u p p + p u p p ln u p. (B.) Since we arrive at u p ln u Q( u p ) = p 4(p ) Q( u, u p ) (B.3) σp 4(p ) Q( u, u p ) + β(σ) p u p p + u p p ln u p. (B.4) By the nonincreasing property of β we then infer, replacing σ by 4(p ) σ p u p ln u σq( u, u p ) + β(σ) p u p p + u p p ln u p (B.5) for σ (0, p 4(p ) σ ]. Part We continue with the assumption Ψ 0. Consider 0 < t 4 σ. Let σ(p) be p a nonnegative continuous function for p such that σ(p) (0, 4(p ) σ ] for p >, which will be chosen later. Then we have u p ln u σ(p)q( u, u p ) + Γ(p) u p p + u p p ln u p (B.6) for each p > and all u W, () L (), where Γ(p) = β(σ(p)). Define the p function p(s) for 0 s < t by Assume that dp ds = p, p(0) =. σ(p) (B.7) p(s) as s t. We also define the function N(s) for 0 s < t by dn ds = Γ(p(s)) σ(s), N(0) = 0 (B.8) (B.9) and set N Γ(p) = lim s t p dp. (B.0) 4

25 For u W, () L () with u 0 we set u s = e sh u for 0 < s < t. By the contraction properties of e sh we have u s W, () L () for all s. If Ψ C () we have for a fixed q > d ds u s q q = q u s s uq s = q Hu s u q s. (B.) Hence d ds u s q q = qq(u s, us q ). (B.) In the general case Ψ L (), this formula follows from the spectral formula for e sh. Using this formula we compute d ds ln(e N(s) u s p(s) ) = d ( N(s) + ) ds p(s) ln u s p(s) p(s) = Γ σ p p σ ln u s p p + ( p u s p p pq(u s, u p s ) + p ) u p s σ ln u s = ( ) σ u s p p u p s ln u s σq(u s, u p s ) Γ u s p p u s p p ln u s p. (B.3) By (B.6) this is nonpositive. Hence e N(s) u s p(s) is nonincreasing, which leads to e sh u p(s) e N(s) f (B.4) for all 0 s < t. By the contraction properties we have e th u p(s) e sh u p(s), whence e th u p(s) e N(s) f (B.5) for all 0 s < t. It follows that e th u e N u. (B.6) This estimate extends to u L () with u 0 by an approximation. For a general u L () we use the pointwise inequality e th u e th f (a consequence of the positivity preserving property) to deduce e th u e th u e N u. (B.7) Now we choose σ(p) = t p. One readily sees that σ(p) (0, p for p. Then p(s) = t t s p(s) as s t. We have for this choice (B.8) 4(p ) σ ] for p > and N = t t 0 β(σ)dσ. (B.9) 5

26 Hence we arrive at for all u L () and 0 < t 4 σ. e th u e τ(t) u (B.0) Part 3 For a general Ψ, we consider Ψ = Ψ min Ψ and denote the corresponding operator and quadratic form by H and Q respectively. We have by (5.4) u ln u dvol σ Q(u) + β(σ) (B.) for all u L () with u =, where β(σ) = β(σ)+σ min Ψ. We apply (B.0) to deduce for 0 < t 4 σ and u L () e t Hu e t R t 0 β(σ)dσ u = e τ(t)+ t 4 min Ψ u. (B.) The desired estimate (5.6) follows. The estimate (5.7) follows from (5.6) in terms of duality. Namely we have for u, v L () ve th u = ue th v e th v u e τ(t) 3t 4 min Ψ v u (B.3) It follows that and then e th u e τ(t) 3t 4 min Ψ u (B.4) e th u e τ( t ) 3t 8 minψ e t Hu e τ( t ) 3t 4 min Ψ u. (B.5) By an approximation, we arrive at (5.7) for all u L (). The estimate (5.7) also follows from the arguments in Part by choosing σ(p) = t t and p(s) =. (In [Z], p t s Zhang essentially used this choice to derive an estimate similar to (5.7).) References [C] [D] [Ga] G. Carron, Inégalités isopérimétriques de Faber-Krahn et conséquences, Actes de la Tables Ronde de Géométrie Différentielle (Luminy, 99), 05-3, Sémin. Congr.,, Soc. ath. France, paris, 996. E. B. Davies, Heat Kernel and Spectral Theory, Cambridge University Press, 989. S. Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque (988),

27 [Ga] [HI] [P] [ST] [Y] [Y] [Y3] S. Gallot, Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, Astérisque (988), 3-9. R. S. Hamilton and J. Isenberg, Quasi-convergence of Ricci flow for a class of metrics, Comm. Anal. Geom. (993), G. Perelman, The netropy formula for the Ricci flow and its geometric applications, Nov. 00, N. Sesum and G. Tian, Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman) and some applications, preprint. R. Ye, Curvature estimates for the Ricci flow I, arxiv:math/ R. Ye, Neumann Type aximum Principles, in preparation. R. Ye, Notes on the logarithmic Sobolev inequality and its application to the Ricci flow, in preparation. [Z] Qi S. Zhang, A uniform Sobolev inequality under Ricci flow, arxiv: v [math.dg], June