Rough metrics, the Kato square root problem, and the continuity of a flow tangent to the Ricci flow

Transcription

1 Rough metrics, the Kato square root problem, and the continuity of a flow tangent to the Ricci flow Lashi Bandara Mathematical Sciences Chalmers University of Technology and University of Gothenburg 22 September 2015 Analyse Harmonique Département de Mathématiques d Orsay Lashi Bandara The continuity of a flow tangent to the Ricci flow 1/30

2 Motivation: the flow of Gigli-Mantegazza Let M be a compact manifold with a smooth metric g. Let g be its Laplacian (on functions) and ρ g (, ) C (R + M M) denote the heat kernel. Lashi Bandara The continuity of a flow tangent to the Ricci flow 2/30

3 Motivation: the flow of Gigli-Mantegazza Let M be a compact manifold with a smooth metric g. Let g be its Laplacian (on functions) and ρ g (, ) C (R + M M) denote the heat kernel. Fix a point x M and a time t > 0, and two tangent vectors u, v T x M. Let ϕ t,x,v L 2 (M) with M ϕ t,x,v dµ g = 0 be the solution to the PDE: div g,y ρ g t (x, y) ϕ t,x,v(y) = d x (ρ g t (x, y))(v), (GMC) Lashi Bandara The continuity of a flow tangent to the Ricci flow 2/30

4 Motivation: the flow of Gigli-Mantegazza Let M be a compact manifold with a smooth metric g. Let g be its Laplacian (on functions) and ρ g (, ) C (R + M M) denote the heat kernel. Fix a point x M and a time t > 0, and two tangent vectors u, v T x M. Let ϕ t,x,v L 2 (M) with M ϕ t,x,v dµ g = 0 be the solution to the PDE: div g,y ρ g t (x, y) ϕ t,x,v(y) = d x (ρ g t (x, y))(v), (GMC) Gigli and Mantegazza in [GM] define g t (x) on tangent vectors u, v T x M by the expression: ˆ g t (x)(u, v) = g(y)( ϕ t,x,u (y), ϕ t,x,v (y)) ρ g t (x, y) dµ g(y). M (GM) Lashi Bandara The continuity of a flow tangent to the Ricci flow 2/30

5 This expression can indeed be checked to define an inner product on T x M. Lashi Bandara The continuity of a flow tangent to the Ricci flow 3/30

6 This expression can indeed be checked to define an inner product on T x M. Moreover, Gigli and Mantegazza show that: t g t ( γ(s), γ(s)) t=0 = 2 Ric g ( γ(s), γ(s)), for almost-every s along g-geodesics γ. Lashi Bandara The continuity of a flow tangent to the Ricci flow 3/30

7 This expression can indeed be checked to define an inner product on T x M. Moreover, Gigli and Mantegazza show that: t g t ( γ(s), γ(s)) t=0 = 2 Ric g ( γ(s), γ(s)), for almost-every s along g-geodesics γ. That is, g t is tangential to the Ricci flow in this weak sense. Lashi Bandara The continuity of a flow tangent to the Ricci flow 3/30

8 This expression can indeed be checked to define an inner product on T x M. Moreover, Gigli and Mantegazza show that: t g t ( γ(s), γ(s)) t=0 = 2 Ric g ( γ(s), γ(s)), for almost-every s along g-geodesics γ. That is, g t is tangential to the Ricci flow in this weak sense. The defining equation (GMC) can be lifted to a distributional equation in Wasserstein space. Lashi Bandara The continuity of a flow tangent to the Ricci flow 3/30

9 This expression can indeed be checked to define an inner product on T x M. Moreover, Gigli and Mantegazza show that: t g t ( γ(s), γ(s)) t=0 = 2 Ric g ( γ(s), γ(s)), for almost-every s along g-geodesics γ. That is, g t is tangential to the Ricci flow in this weak sense. The defining equation (GMC) can be lifted to a distributional equation in Wasserstein space. Using the induced heat flow, we obtain a time evolving family of distance metrics d t starting with the initial metric d 0 = d g, the induced distance from g. Lashi Bandara The continuity of a flow tangent to the Ricci flow 3/30

10 This expression can indeed be checked to define an inner product on T x M. Moreover, Gigli and Mantegazza show that: t g t ( γ(s), γ(s)) t=0 = 2 Ric g ( γ(s), γ(s)), for almost-every s along g-geodesics γ. That is, g t is tangential to the Ricci flow in this weak sense. The defining equation (GMC) can be lifted to a distributional equation in Wasserstein space. Using the induced heat flow, we obtain a time evolving family of distance metrics d t starting with the initial metric d 0 = d g, the induced distance from g. Indeed, d t is induced from the metrics g t defined by (GM). Lashi Bandara The continuity of a flow tangent to the Ricci flow 3/30

11 In fact, the fact that (GMC) can be given meaning in Wasserstein space means exactly that the flow of distance metrics d t can be defined for an RCD-space (X, d, µ) (a measure metric space with a notion of Ricci curvature bounded from below and with a Hilbertian Sobolev space). Lashi Bandara The continuity of a flow tangent to the Ricci flow 4/30

12 In fact, the fact that (GMC) can be given meaning in Wasserstein space means exactly that the flow of distance metrics d t can be defined for an RCD-space (X, d, µ) (a measure metric space with a notion of Ricci curvature bounded from below and with a Hilbertian Sobolev space). In particular, this allows us to flow spaces containing singularities. Lashi Bandara The continuity of a flow tangent to the Ricci flow 4/30

13 In fact, the fact that (GMC) can be given meaning in Wasserstein space means exactly that the flow of distance metrics d t can be defined for an RCD-space (X, d, µ) (a measure metric space with a notion of Ricci curvature bounded from below and with a Hilbertian Sobolev space). In particular, this allows us to flow spaces containing singularities. Given that there are few tools to consider regularity questions in the RCD setting, we consider this problem on a smooth manifold M but with low-regularity metrics. Lashi Bandara The continuity of a flow tangent to the Ricci flow 4/30

14 Rough metrics Assume that M is a manifold (possibly noncompact). Definition (Rough metric) Let g be a (2, 0) symmetric tensor field with measurable coefficients and that for each x M, there is some chart (U, ψ) near x and a constant C 1 such that C 1 u ψ δ(y) u g(y) C u ψ δ(y), for almost-every y U and where δ is the Euclidean metric in ψ(u). Then we say that g is a rough metric, and such a chart (U, ψ) is said to satisfy the local comparability condition. Lashi Bandara The continuity of a flow tangent to the Ricci flow 5/30

15 Induced measure Define µ g for a rough metric g by writing dµ g (x) = det g(x) dl (x) inside charts satisfying the local comparability condition and then patching them together via a partition of unity. Lashi Bandara The continuity of a flow tangent to the Ricci flow 6/30

16 Induced measure Define µ g for a rough metric g by writing dµ g (x) = det g(x) dl (x) inside charts satisfying the local comparability condition and then patching them together via a partition of unity. This measure is Borel-regular and finite on compact sets. It is unknown whether they are generally Radon. However, if M is compact, then it is. Lashi Bandara The continuity of a flow tangent to the Ricci flow 6/30

17 Induced measure Define µ g for a rough metric g by writing dµ g (x) = det g(x) dl (x) inside charts satisfying the local comparability condition and then patching them together via a partition of unity. This measure is Borel-regular and finite on compact sets. It is unknown whether they are generally Radon. However, if M is compact, then it is. Moreover, L p theory exists (trivial) and on C L 2 is a closable, densely-defined operator which gives Sobolev spaces W 1,2 (M) and W 1,2 0 (M). Lashi Bandara The continuity of a flow tangent to the Ricci flow 6/30

18 Metric perturbations Definition We say that two rough metrics g and g are C-close if C 1 u g(x) u g(x) C u g(x) for almost-every x M where C 1. Two such metrics are said to be C-close everywhere if this inequality holds for every x M. Lashi Bandara The continuity of a flow tangent to the Ricci flow 7/30

19 Metric perturbations Definition We say that two rough metrics g and g are C-close if C 1 u g(x) u g(x) C u g(x) for almost-every x M where C 1. Two such metrics are said to be C-close everywhere if this inequality holds for every x M. Note: on a compact manifold, there is always a C-close smooth metric g given a rough metric g. Lashi Bandara The continuity of a flow tangent to the Ricci flow 7/30

20 Proposition Let g and g be two rough metrics that are C-close. Then, there exists B Γ(T M TM) such that it is symmetric, almost-everywhere positive and invertible, and g x (B(x)u, v) = g x (u, v) for almost-every x M. Furthermore, for almost-every x M, C 2 u g(x) B(x)u g(x) C 2 u g(x), and the same inequality with g and g interchanged. If g C k and g C l (with k, l 0), then the properties of B are valid for all x M and B C min{k,l} (T M TM). Lashi Bandara The continuity of a flow tangent to the Ricci flow 8/30

21 The measure µ g (x) = θ(x) dµ g (x), where θ(x) = det B(x). Lashi Bandara The continuity of a flow tangent to the Ricci flow 9/30

22 The measure µ g (x) = θ(x) dµ g (x), where θ(x) = det B(x). Consequently, (i) whenever p [1, ), L p (T (r,s) M, g) = L p (T (r,s) M, g) with ( ) C r+s+ n 2p u p, g u p,g C r+s+ n 2p u p, g, Lashi Bandara The continuity of a flow tangent to the Ricci flow 9/30

23 The measure µ g (x) = θ(x) dµ g (x), where θ(x) = det B(x). Consequently, (i) whenever p [1, ), L p (T (r,s) M, g) = L p (T (r,s) M, g) with ( ) C r+s+ n 2p u p, g u p,g C r+s+ n 2p u p, g, (ii) for p =, L (T (r,s) M, g) = L (T (r,s) M, g) with C (r+s) u, g u,g C r+s u, g, Lashi Bandara The continuity of a flow tangent to the Ricci flow 9/30

24 (iii) the Sobolev spaces W 1,p (M, g) = W 1,p (M, g) and (M, g) = W1,p(M, g) with W 1,p 0 0 ( ) C 1+ n 2p u W 1,p, g u W 1,p,g C 1+ n 2p u W 1,p, g, Lashi Bandara The continuity of a flow tangent to the Ricci flow 10/30

25 (iii) the Sobolev spaces W 1,p (M, g) = W 1,p (M, g) and (M, g) = W1,p(M, g) with W 1,p 0 0 ( ) C 1+ n 2p u W 1,p, g u W 1,p,g C 1+ n 2p u W 1,p, g, (v) the divergence operators satisfy div D,g = θ 1 div D, g θb and div N,g = θ 1 div N, g θb. Lashi Bandara The continuity of a flow tangent to the Ricci flow 10/30

26 (iii) the Sobolev spaces W 1,p (M, g) = W 1,p (M, g) and (M, g) = W1,p(M, g) with W 1,p 0 0 ( ) C 1+ n 2p u W 1,p, g u W 1,p,g C 1+ n 2p u W 1,p, g, (v) the divergence operators satisfy div D,g = θ 1 div D, g θb and div N,g = θ 1 div N, g θb. Note: Rough metrics are natural geometric invariances of the Kato square root problem. See [B2]. Lashi Bandara The continuity of a flow tangent to the Ricci flow 10/30

27 Assume now that M is compact. Lashi Bandara The continuity of a flow tangent to the Ricci flow 11/30

28 Assume now that M is compact. Let g be a rough metric and g a C-close smooth metric to g for some C 1. Lashi Bandara The continuity of a flow tangent to the Ricci flow 11/30

29 Assume now that M is compact. Let g be a rough metric and g a C-close smooth metric to g for some C 1. It makes sense to consider (GMC) in this context, provided we have the existence of a sufficiently good heat kernel. Lashi Bandara The continuity of a flow tangent to the Ricci flow 11/30

30 Assume now that M is compact. Let g be a rough metric and g a C-close smooth metric to g for some C 1. It makes sense to consider (GMC) in this context, provided we have the existence of a sufficiently good heat kernel. Solving (GMC) is equivalent to solving div g,y ρ g t (x, y)bθ ϕ t,x,v = θ d x (ρ g t (x, y))(v), (GMC ) where g(bu, v) = g(u, v). Lashi Bandara The continuity of a flow tangent to the Ricci flow 11/30

31 Assume now that M is compact. Let g be a rough metric and g a C-close smooth metric to g for some C 1. It makes sense to consider (GMC) in this context, provided we have the existence of a sufficiently good heat kernel. Solving (GMC) is equivalent to solving div g,y ρ g t (x, y)bθ ϕ t,x,v = θ d x (ρ g t (x, y))(v), (GMC ) where g(bu, v) = g(u, v). Concern: regularity of the metric x g t (x)(u, v) = ρ g t (x, ) ϕ t,x,u, ϕ t,x,v L 2 (g). Lashi Bandara The continuity of a flow tangent to the Ricci flow 11/30

32 Theorem (B., Lakzian, Munn ([BLM], 2015)) Let M be a smooth, compact manifold and g a rough metric. Let N M be an open set. Lashi Bandara The continuity of a flow tangent to the Ricci flow 12/30

33 Theorem (B., Lakzian, Munn ([BLM], 2015)) Let M be a smooth, compact manifold and g a rough metric. Let N M be an open set. (i) If the the heat kernel (x, y) ρ g t (x, y) C0,1 (M 2 ) and improves to (x, y) ρ g t (x, y) Ck (N 2 ) where k 2. Then, for t > 0, g t is a Riemannian metric on N of regularity C k 2,1. Lashi Bandara The continuity of a flow tangent to the Ricci flow 12/30

34 Theorem (B., Lakzian, Munn ([BLM], 2015)) Let M be a smooth, compact manifold and g a rough metric. Let N M be an open set. (i) If the the heat kernel (x, y) ρ g t (x, y) C0,1 (M 2 ) and improves to (x, y) ρ g t (x, y) Ck (N 2 ) where k 2. Then, for t > 0, g t is a Riemannian metric on N of regularity C k 2,1. (ii) If the heat kernel (x, y) ρ g t (x, y) C1 (M 2 ) and (x, y) ρ g t (x, y) Ck (N 2 ) where k 1. Then, for t > 0, g t is a Riemannian metric on N of regularity C k 1. Lashi Bandara The continuity of a flow tangent to the Ricci flow 12/30

35 Standing question: What happens if we only assume that (x, y) ρ g t (x, y) C1 (N 2 ) (i.e., no C 0,1 or C 1 assumptions on global regularity). Lashi Bandara The continuity of a flow tangent to the Ricci flow 13/30

36 Standing question: What happens if we only assume that (x, y) ρ g t (x, y) C1 (N 2 ) (i.e., no C 0,1 or C 1 assumptions on global regularity). Is it true then that x g t (x) C 0? Lashi Bandara The continuity of a flow tangent to the Ricci flow 13/30

37 Standing question: What happens if we only assume that (x, y) ρ g t (x, y) C1 (N 2 ) (i.e., no C 0,1 or C 1 assumptions on global regularity). Is it true then that x g t (x) C 0? Theorem (B. ([B], 2015)) Let M be a smooth, compact manifold, and N M, an open set. Suppose that g is a rough metric and that ρ g t C1 (N 2 ). Then, g t as defined by (GM) exists on N and it is continuous. Lashi Bandara The continuity of a flow tangent to the Ricci flow 13/30

38 The equation (GMC) is a specific case of pointwise linear problems of the form: L x u x = η x (PE) for suitable source data η x L 2 (M) and where L x = div A x is a family of divergence form operators. Lashi Bandara The continuity of a flow tangent to the Ricci flow 14/30

39 The equation (GMC) is a specific case of pointwise linear problems of the form: L x u x = η x (PE) for suitable source data η x L 2 (M) and where L x = div A x is a family of divergence form operators. Theorem (B. ([B], 2015)) Let M be a smooth manifold and g a smooth metric. At x M suppose that x A x are real, symmetric, elliptic, bounded measurable coefficients that are L -continuous at x, and that x η x is L 2 -continuous at x. If x u x solves (PE) at x with M η x dµ g = 0, then x u x is L 2 -continuous at x. Lashi Bandara The continuity of a flow tangent to the Ricci flow 14/30

40 Representation of solutions to the PDE The equation (PE) can be further reduced to studying elliptic problems of the form L A u = div g A u = f, (E) for suitable source data f L 2 (M), where the coefficients A are symmetric, bounded, measurable and for which there exists a κ > 0 satisfying Au, u κ u 2. Lashi Bandara The continuity of a flow tangent to the Ricci flow 15/30

41 Representation of solutions to the PDE The equation (PE) can be further reduced to studying elliptic problems of the form L A u = div g A u = f, (E) for suitable source data f L 2 (M), where the coefficients A are symmetric, bounded, measurable and for which there exists a κ > 0 satisfying Au, u κ u 2. By the operator theory of self-adjoint operators, we obtain that L 2 (M) = N (L A ) R(L A ). Lashi Bandara The continuity of a flow tangent to the Ricci flow 15/30

42 Moreover, N (L A ) = N ( ). Lashi Bandara The continuity of a flow tangent to the Ricci flow 16/30

43 Moreover, N (L A ) = N ( ). Since (M, g) is smooth and compact, there is a Poincaré inequality, and since A are bounded below, R = R(L A ) = R( L A ), where { ˆ } R = u L 2 (M) : u dµ g = 0. M Lashi Bandara The continuity of a flow tangent to the Ricci flow 16/30

44 Moreover, N (L A ) = N ( ). Since (M, g) is smooth and compact, there is a Poincaré inequality, and since A are bounded below, R = R(L A ) = R( L A ), where { ˆ } R = u L 2 (M) : u dµ g = 0. M Also, the embedding E : W 1,2 (M) L 2 (M) is compact, which implies that the spectrum of L A is discrete. Lashi Bandara The continuity of a flow tangent to the Ricci flow 16/30

45 Moreover, N (L A ) = N ( ). Since (M, g) is smooth and compact, there is a Poincaré inequality, and since A are bounded below, R = R(L A ) = R( L A ), where { ˆ } R = u L 2 (M) : u dµ g = 0. M Also, the embedding E : W 1,2 (M) L 2 (M) is compact, which implies that the spectrum of L A is discrete. The Poincaré inequality implies a spectral gap between the zero and the first-nonzero eigenvalues. Lashi Bandara The continuity of a flow tangent to the Ricci flow 16/30

46 Moreover, N (L A ) = N ( ). Since (M, g) is smooth and compact, there is a Poincaré inequality, and since A are bounded below, R = R(L A ) = R( L A ), where { ˆ } R = u L 2 (M) : u dµ g = 0. M Also, the embedding E : W 1,2 (M) L 2 (M) is compact, which implies that the spectrum of L A is discrete. The Poincaré inequality implies a spectral gap between the zero and the first-nonzero eigenvalues. Then, for f R, u = L 1 A f is a solution to (E) satisfying M u dµ g = 0. Lashi Bandara The continuity of a flow tangent to the Ricci flow 16/30

47 Back to the pointwise elliptic linear equation Suppose that A x u, u κ x u 2, for u L 2 (T M). Lashi Bandara The continuity of a flow tangent to the Ricci flow 17/30

48 Back to the pointwise elliptic linear equation Suppose that A x u, u κ x u 2, for u L 2 (T M). Let T x = L x = div A x, and let u x, u y L 2 (M) such that M u x dµ g = M u y dµ g = 0. Lashi Bandara The continuity of a flow tangent to the Ricci flow 17/30

49 Back to the pointwise elliptic linear equation Suppose that A x u, u κ x u 2, for u L 2 (T M). Let T x = L x = div A x, and let u x, u y L 2 (M) such that M u x dµ g = M u y dµ g = 0. Then, L 1 x u x L 1 u y = T 1 where v x = Tx 1 u x and v y = Ty 1 u y. y x v x Ty 1 v y Lashi Bandara The continuity of a flow tangent to the Ricci flow 17/30

50 Back to the pointwise elliptic linear equation Suppose that A x u, u κ x u 2, for u L 2 (T M). Let T x = L x = div A x, and let u x, u y L 2 (M) such that M u x dµ g = M u y dµ g = 0. Then, L 1 x u x L 1 u y = T 1 y x v x Ty 1 v y where v x = Tx 1 u x and v y = Ty 1 u y. To prove L 2 continuity, it suffices to show that T 1 x v x Ty 1 v y A x A y v x + v x v y. Lashi Bandara The continuity of a flow tangent to the Ricci flow 17/30

51 Also, T 1 x v x T 1 v y (T 1 y x (T 1 x T 1 y )v x + T 1 (v x v y ) Ty 1 )v x + (Ty 1 + Tx 1 (v x v y ). y Tx 1 )(v x v y ) Lashi Bandara The continuity of a flow tangent to the Ricci flow 18/30

52 Also, T 1 x v x T 1 v y (T 1 y x (T 1 x T 1 y So, for u L 2 (M) with M u dµ g = 0, T 1 x u T 1 y )v x + T 1 (v x v y ) Ty 1 )v x + (Ty 1 + Tx 1 (v x v y ). u = T 1 T y T 1 u T 1 T x T 1 u x = T 1 x y x y y Tx 1 )(v x v y ) (T y T x )T 1 u (T y T x )T 1 u y y Lashi Bandara The continuity of a flow tangent to the Ricci flow 18/30

53 Also, T 1 x v x T 1 v y (T 1 y x (T 1 x T 1 y So, for u L 2 (M) with M u dµ g = 0, T 1 x u T 1 y Thus, it suffices to show )v x + T 1 (v x v y ) Ty 1 )v x + (Ty 1 + Tx 1 (v x v y ). u = T 1 T y T 1 u T 1 T x T 1 u x = T 1 x y x y y Tx 1 )(v x v y ) (T y T x )T 1 u (T y T x )T 1 u L x u L y u A x A y u. y y Lashi Bandara The continuity of a flow tangent to the Ricci flow 18/30

54 Also, T 1 x v x T 1 v y (T 1 y x (T 1 x T 1 y So, for u L 2 (M) with M u dµ g = 0, T 1 x u T 1 y Thus, it suffices to show )v x + T 1 (v x v y ) Ty 1 )v x + (Ty 1 + Tx 1 (v x v y ). u = T 1 T y T 1 u T 1 T x T 1 u x = T 1 x y x y y Tx 1 )(v x v y ) (T y T x )T 1 u (T y T x )T 1 u L x u L y u A x A y u. Such an estimate follows from holomorphic dependency of the functional calculus if we are able to prove a homogeneous Kato square root estimate. y y Lashi Bandara The continuity of a flow tangent to the Ricci flow 18/30

55 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), Lashi Bandara The continuity of a flow tangent to the Ricci flow 19/30

56 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), (H2) B 1, B 2 L(H ) and there exist κ 1, κ 2 > 0 satisfying the accretivity conditions Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2, for u R(Γ ) and v R(Γ), and Lashi Bandara The continuity of a flow tangent to the Ricci flow 19/30

57 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), (H2) B 1, B 2 L(H ) and there exist κ 1, κ 2 > 0 satisfying the accretivity conditions Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2, for u R(Γ ) and v R(Γ), and (H3) B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Lashi Bandara The continuity of a flow tangent to the Ricci flow 19/30

58 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), (H2) B 1, B 2 L(H ) and there exist κ 1, κ 2 > 0 satisfying the accretivity conditions Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2, for u R(Γ ) and v R(Γ), and (H3) B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Let us now define Π B = Γ + B 1 Γ B 2 with domain D(Π B ) = D(Γ) D(B 1 Γ B 2 ). Lashi Bandara The continuity of a flow tangent to the Ricci flow 19/30

59 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), (H2) B 1, B 2 L(H ) and there exist κ 1, κ 2 > 0 satisfying the accretivity conditions Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2, for u R(Γ ) and v R(Γ), and (H3) B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Let us now define Π B = Γ + B 1 Γ B 2 with domain D(Π B ) = D(Γ) D(B 1 Γ B 2 ). Π B is an ω-bisectorial operator with ω [0, π/2). Lashi Bandara The continuity of a flow tangent to the Ricci flow 19/30

60 Quadratic estimates To say that Π B satisfies quadratic estimates means that ˆ 0 tπ B (I + t 2 Π 2 B) 1 u 2 dt t u 2, (Q) for all u R(Π B ). Lashi Bandara The continuity of a flow tangent to the Ricci flow 20/30

61 Quadratic estimates To say that Π B satisfies quadratic estimates means that ˆ 0 tπ B (I + t 2 Π 2 B) 1 u 2 dt t u 2, (Q) for all u R(Π B ). This implies that D( Π 2 B ) = D(Π B) = D(Γ) D(Γ B 2 ) Π 2 B u Π Bu Γu + Γ B 2 u Lashi Bandara The continuity of a flow tangent to the Ricci flow 20/30

62 More importantly, for coefficients A 1, A 2 L(H ) satisfying (i) A i η i < κ i, (ii) A 1 A 2 R(Γ), B 1 A 2 R(Γ), A 1 B 2 R(Γ) N (Γ), and (iii) A 2 A 1 R(Γ ), B 2 A 1 R(Γ ), A 2 B 1 R(Γ ) N (Γ ), we obtain that for an appropriately chosen µ < π/2, and for all bounded holomorphic functions f in an open bisector containing the closed ω-bisector, f(π B ) f(π B+A ) ( A 1 + A 2 ) f. (Hol) Lashi Bandara The continuity of a flow tangent to the Ricci flow 21/30

63 More importantly, for coefficients A 1, A 2 L(H ) satisfying (i) A i η i < κ i, (ii) A 1 A 2 R(Γ), B 1 A 2 R(Γ), A 1 B 2 R(Γ) N (Γ), and (iii) A 2 A 1 R(Γ ), B 2 A 1 R(Γ ), A 2 B 1 R(Γ ) N (Γ ), we obtain that for an appropriately chosen µ < π/2, and for all bounded holomorphic functions f in an open bisector containing the closed ω-bisector, f(π B ) f(π B+A ) ( A 1 + A 2 ) f. (Hol) This framework and connections to the Kato square root problem can be found in their paper [AKMc]. Lashi Bandara The continuity of a flow tangent to the Ricci flow 21/30

64 More importantly, for coefficients A 1, A 2 L(H ) satisfying (i) A i η i < κ i, (ii) A 1 A 2 R(Γ), B 1 A 2 R(Γ), A 1 B 2 R(Γ) N (Γ), and (iii) A 2 A 1 R(Γ ), B 2 A 1 R(Γ ), A 2 B 1 R(Γ ) N (Γ ), we obtain that for an appropriately chosen µ < π/2, and for all bounded holomorphic functions f in an open bisector containing the closed ω-bisector, f(π B ) f(π B+A ) ( A 1 + A 2 ) f. (Hol) This framework and connections to the Kato square root problem can be found in their paper [AKMc]. This is a first-order reformulation Kato square root problem resolved by Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian in [AHLMcT]. Lashi Bandara The continuity of a flow tangent to the Ricci flow 21/30

65 The Kato square root problem on manifolds Let H = L 2 (M) L 2 (M) L 2 (T M), and set S = (I, ) Lashi Bandara The continuity of a flow tangent to the Ricci flow 22/30

66 The Kato square root problem on manifolds Let H = L 2 (M) L 2 (M) L 2 (T M), and set S = (I, ) Set Γ = ( ) 0 0, and s 0 ( 0 S 0 0 ). Lashi Bandara The continuity of a flow tangent to the Ricci flow 22/30

67 The Kato square root problem on manifolds Let H = L 2 (M) L 2 (M) L 2 (T M), and set S = (I, ) Set Γ = ( ) 0 0, and s 0 ( 0 S 0 0 ). B 1 = ( ) a 0, and B = for a L (M) and A L ((M C) T M). ( ) A Lashi Bandara The continuity of a flow tangent to the Ricci flow 22/30

68 Theorem (B., McIntosh ([BMc], 2012)) Let (M, g) be a smooth, complete Riemannian manifold with Ric C and inj(m) κ > 0. Suppose that the following ellipticity condition holds: there exist κ 1, κ 2 > 0 such that Re au, u κ 1 u 2 and Re( A 11 v, v + A 10 v, v + A 01 v, v + A 00 v, v ) κ 2 v 2 W 1,2 for all u L 2 (M) and v W 1,2 (M). Let D A u = a div A 11 u a div A 10 u + aa 01 u + aa 00 u. Then, the quadratic estimates (Q) are satisfied, D( D A ) = D( ) = W 1,2 (M) with D A u u + u = u W 1,2 for all u W 1,2 (M), and D A u D B u A B u W 1,2, whenever b, B are coefficients that satisfy accretivity assumptions with η i < κ i Lashi Bandara The continuity of a flow tangent to the Ricci flow 23/30

69 Every smooth compact Riemannian manifold (M, g) satisfies the geometric assumptions: it is complete, Ric C, and there exists κ > 0 such that inj(m, g) κ. Lashi Bandara The continuity of a flow tangent to the Ricci flow 24/30

70 Every smooth compact Riemannian manifold (M, g) satisfies the geometric assumptions: it is complete, Ric C, and there exists κ > 0 such that inj(m, g) κ. We want L A = D A, but if we take A 10, A 10 and A 00 to be 0, we lose accretivity. Lashi Bandara The continuity of a flow tangent to the Ricci flow 24/30

71 Every smooth compact Riemannian manifold (M, g) satisfies the geometric assumptions: it is complete, Ric C, and there exists κ > 0 such that inj(m, g) κ. We want L A = D A, but if we take A 10, A 10 and A 00 to be 0, we lose accretivity. The norm in the Lipschitz perturbation estimate is a W 1,2 norm, but we need L 2 norms. Lashi Bandara The continuity of a flow tangent to the Ricci flow 24/30

72 Every smooth compact Riemannian manifold (M, g) satisfies the geometric assumptions: it is complete, Ric C, and there exists κ > 0 such that inj(m, g) κ. We want L A = D A, but if we take A 10, A 10 and A 00 to be 0, we lose accretivity. The norm in the Lipschitz perturbation estimate is a W 1,2 norm, but we need L 2 norms. The estimate of central importance here is the following coercivity estimate: u Πu, for u D(Π) R(Π). Lashi Bandara The continuity of a flow tangent to the Ricci flow 24/30

73 Every smooth compact Riemannian manifold (M, g) satisfies the geometric assumptions: it is complete, Ric C, and there exists κ > 0 such that inj(m, g) κ. We want L A = D A, but if we take A 10, A 10 and A 00 to be 0, we lose accretivity. The norm in the Lipschitz perturbation estimate is a W 1,2 norm, but we need L 2 norms. The estimate of central importance here is the following coercivity estimate: u Πu, for u D(Π) R(Π). This is almost trivial for the inhomogeneous problem. Lashi Bandara The continuity of a flow tangent to the Ricci flow 24/30

74 The homogeneous Kato square root problem on compact manifolds Let H = L 2 (M) L 2 (T M). Lashi Bandara The continuity of a flow tangent to the Ricci flow 25/30

75 The homogeneous Kato square root problem on compact manifolds Let H = L 2 (M) L 2 (T M). Set Γ = ( ) 0 0, and 0 ( ) 0 div. 0 0 Lashi Bandara The continuity of a flow tangent to the Ricci flow 25/30

76 The homogeneous Kato square root problem on compact manifolds Let H = L 2 (M) L 2 (T M). Set Γ = ( ) 0 0, and 0 ( ) 0 div. 0 0 B 1 = ( ) a 0, and B = for a L (M) and A L (T M). ( ) A Lashi Bandara The continuity of a flow tangent to the Ricci flow 25/30

77 By self-adjointness for Π and, if the coefficients satisfy (H1)-(H3) by bi-sectoriality, H = N (Π) R(Π) = N (Π B ) R(Π B ). Lashi Bandara The continuity of a flow tangent to the Ricci flow 26/30

78 By self-adjointness for Π and, if the coefficients satisfy (H1)-(H3) by bi-sectoriality, H = N (Π) R(Π) = N (Π B ) R(Π B ). Thus, we have that L 2 (M) = N ( ) R(div) and L 2 (T M) = N (div) R( ). Lashi Bandara The continuity of a flow tangent to the Ricci flow 26/30

79 By self-adjointness for Π and, if the coefficients satisfy (H1)-(H3) by bi-sectoriality, H = N (Π) R(Π) = N (Π B ) R(Π B ). Thus, we have that L 2 (M) = N ( ) R(div) and L 2 (T M) = N (div) R( ). Moreover, R(div) = { ˆ } u L 2 (M) : u dµ g = 0 = R. M Lashi Bandara The continuity of a flow tangent to the Ricci flow 26/30

80 Now, let u R(Π) D(Π). So u = (u 1, u 2 ), and Πu = u 1 + div u 2. Lashi Bandara The continuity of a flow tangent to the Ricci flow 27/30

81 Now, let u R(Π) D(Π). So u = (u 1, u 2 ), and Πu = u 1 + div u 2. Poincaré inequality then gives us that u 1 C u 1, since u 1 R(div). Lashi Bandara The continuity of a flow tangent to the Ricci flow 27/30

82 Now, let u R(Π) D(Π). So u = (u 1, u 2 ), and Πu = u 1 + div u 2. Poincaré inequality then gives us that u 1 C u 1, since u 1 R(div). Now, u 2 = v 2, for some v 2 D(Π). So, div u 2 = v 2 = v 2 C v 2 = C v 2 = C u 2. Lashi Bandara The continuity of a flow tangent to the Ricci flow 27/30

83 Now, let u R(Π) D(Π). So u = (u 1, u 2 ), and Πu = u 1 + div u 2. Poincaré inequality then gives us that u 1 C u 1, since u 1 R(div). Now, u 2 = v 2, for some v 2 D(Π). So, div u 2 = v 2 = v 2 C v 2 = C v 2 = C u 2. That is, Πu C u. Lashi Bandara The continuity of a flow tangent to the Ricci flow 27/30

84 Theorem (B., ([B], 2015)) On a compact manifold M with a smooth metric g, the operator Π B admits a bounded functional calculus. In particular, D( b div B ) = W 1,2 (M) and b div B u u. Moreover, whenever b < η 1 and B < η 2, where η i < κ i, we have the following Lipschitz estimate b div B u (b + b) div(b + B) u ( b + B ) u whenever u W 1,2 (M). The implicit constant depends on b, B and η i. Lashi Bandara The continuity of a flow tangent to the Ricci flow 28/30

85 Corollary Fix x M and u W 1,2 (M). If A x A y ζ < κ x, then for u W 1,2 (M), L x u L y u A x A y u. The implicit constant depends on ζ and A x. Lashi Bandara The continuity of a flow tangent to the Ricci flow 29/30

86 Corollary Fix x M and u W 1,2 (M). If A x A y ζ < κ x, then for u W 1,2 (M), L x u L y u A x A y u. The implicit constant depends on ζ and A x. Corollary Fix x M and suppose that A x A y ζ < κ x. Then, L 1 x η x L 1 y η y A x A y η x + η x η y, whenever η x, η y L 2 (M) satisfies M η x dµ g = M η y dµ g = 0. The implicit constant depends on ζ, κ x, and A x. Lashi Bandara The continuity of a flow tangent to the Ricci flow 29/30

87 References [AHLMcT] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philippe. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on R n, Ann. of Math. (2) 156 (2002), no. 2, [AKMc] Andreas Axelsson, Stephen Keith, and Alan McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), no. 3, MR (2007k:58029) [B2] L. Bandara, Rough metrics on manifolds and quadratic estimates, ArXiv e-prints (2014). [B] L. Bandara, Continuity of solutions to space-varying pointwise linear elliptic equations, ArXiv e-prints (2015). [BLM] L. Bandara, S. Lakzian, and M. Munn, Geometric singularities and a flow tangent to the Ricci flow, ArXiv e-prints (2015). [BMc] Lashi Bandara and Alan McIntosh, The Kato Square Root Problem on Vector Bundles with Generalised Bounded Geometry, The Journal of Geometric Analysis (2015), [GM] Nicola Gigli and Carlo Mantegazza, A flow tangent to the ricci flow via heat kernels and mass transport, Advances in Mathematics 250 (2014), Lashi Bandara The continuity of a flow tangent to the Ricci flow 30/30