Geometric Analysis on Riemannian and Metric Spaces

Transcription

1 RIMS Project 2016: Differential Geometry and Geometric Analysis Conference Geometric Analysis on Riemannian and Metric Spaces September 5 (Mon) 9 (Fri), 2016 Room 420, Research Institute for Mathematical Sciences, Kyoto University Sep 5 (Mon) Program 10:00 11:00 Kazumasa Kuwada (Tokyo Institute of Technology) Monotonicity and Rigidity of the W-entropy on RCD (0, N) spaces 11:20 12:20 Xiang-Dong Li (Chinese Academy of Sciences) W -entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds 14:00 15:00 Qing-Ming Cheng (Fukuoka) Geometry of complete λ-hypersurfaces 15:20 16:20 Shouhei Honda (Tohoku) Spectral convergence under bounded Ricci curvature Sep 6 (Tue) 10:00 11:00 William Wylie (Syracuse) Weighted Ricci curvature with effective dimension one on smooth manifolds 11:20 12:20 Homare Tadano (Osaka) Some Myers Type Theorems and Hitchin-Thorpe Inequalities for Shrinking Ricci Solitons 14:00 15:00 Masashi Ishida (Tohoku) Obstructions to the existence of non-singular solutions of the normalized Ricci flow on four-manifolds 15:20 16:20 Andrea Mondino (ETH Zurich) On the fundamental group of non smooth spaces with Ricci curvature bounded from below 16:40 17:40 Karl-Theodor Sturm (Bonn) Heat flow on time-dependent mm-spaces and super Ricci flows

2 Sep 7 (Wed) 10:00 11:00 Ovidiu Munteanu (Connecticut) Ricci Solitons 11:20 12:20 Shin-ichi Ohta (Kyoto) Nonlinear geometric analysis on Finsler manifolds: Gaussian isoperimetric inequality Sep 8 (Thu) 10:00 11:00 Christian Ketterer (Freiburg) Lagrangian calculus for non-symmetric diffusion operators 11:20 12:20 Yu Kitabeppu (Kyoto) On RCD spaces with Bishop-type inequalities 14:00 15:00 Kei Kondo (Yamaguchi) Non-smooth analysis and differentiable sphere theorems 15:20 16:20 Ayato Mitsuishi (Gakushuin) Orientabilities and fundamental classes of Alexandrov spaces and its applications 16:40 17:40 Nicola Gigli (SISSA, Trieste) Nonsmooth differential geometry Sep 9 (Fri) 10:00 11:00 Takao Yamaguchi (Kyoto) Inradius collapsed manifolds 11:20 12:20 Xiaochun Rong (CNU and Rutgers) Quantitative volume space form rigidity with Ricci curvature bounded below Organizers Takao Yamaguchi Takashi Shioya Shin-ichi Ohta Shouhei Honda Takumi Yokota (Kyoto) (Tohoku) (Kyoto) (Tohoku) (RIMS, Kyoto)

3 Abstracts Qing-Ming Cheng (Fukuoka) Geometry of complete λ-hypersurfaces In this talk, we give a definition of λ-hypersurface, which are critical points of the weighted area functional for the weighted volume-preserving variations. We construct examples of complete λ-hypersurfaces. Especially, we prove that there exists embedded compact λ-torus. Furthermore, the area growth on complete λ-hypersurfaces is discussed. Nicola Gigli (SISSA, Trieste) Nonsmooth differential geometry Aim of the talk is to give an overview on some recent results concerning the possibility of developing differential calculus both on abstract metric measure spaces and on spaces with Ricci curvature bounded from below. Specifically, building on top of some ideas of Weaver, I shall describe in which sense general metric measure spaces possess a weak first order differential structure. The construction is based on the notion of L 2 -normed L -module: these are Banach spaces which are also modules over the ring of L -function and which possess a pointwise norm, i.e. a map defined in them and taking values in L 2 compatible, in the appropriate sense, with both the Banach and the L -module structure. The key example of such structure is the space of L 2 sections of a normed vector bundle on a manifold and the advantage of using this approach in the nonsmooth setting - reminiscent of Serre- Swan theorem - is that one can speak of (co)vector fields without having to specify the (co)tangent space at every point. Such a general setup will then by applied in the setting of spaces with Ricci curvature bounded from below. Having a weak curvature bound reflects in the possibility of defining a second order calculus, thus allowing to speak about Hessians, covariant and exterior derivatives as well as a Ricci curvature tensor. Shouhei Honda (Tohoku) Spectral convergence under bounded Ricci curvature For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding s spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov- Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of

4 Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli s one and Lott s Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott s question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. This talk is based on arxiv: Masashi Ishida (Tohoku) Obstructions to the existence of non-singular solutions of the normalized Ricci flow on four-manifolds A solution of the normalized Ricci flow is called non-singular if the solution exists for all time and the Riemannian curvature tensor is uniformly bounded. In 1999, R.Hamilton introduced this notion as a nice class of solutions and classified 3-dimensional non-singular solutions. In this talk, we shall discuss obstructions to the existence of 4-dimensional non-singular solutions and give applications of the obstructions. Christian Ketterer (Freiburg) Lagrangian calculus for non-symmetric diffusion operators In this talk I present an optimal transport characterization of lower bounds for the Bakry-Emery Ricci tensor of non-symmetric diffusion operators on generalized smooth metric measure spaces. Consequences are Bishop-Gromov-type estimates, precompactness under measured Gromov-Hausdorff convergence, and a Bonnet-Myers theorem that generalizes previous results by Kuwada. For smooth Riemannian manifolds we derive an evolution variational inequality and contraction estimates for the dual semi-group acting on probability measures. Another theorem of Kuwada yields Bakry- Emery gradient estimates. Yu Kitabeppu (Kyoto) On RCD spaces with Bishop-type inequalities It is known that all tangent cones at any points of non-collapsing Ricci limit spaces are metric cones. On the other hand, there are collapsing Ricci limit spaces that have a point whose tangent cones are not metric cone. The family of metric measure spaces with RCD conditions includes all Ricci limit spaces, so that we cannot expect that generic tangent cones are metric cones for RCD settings. In my talk, I introduce the new Bishop-type inequality, and I show that tangent cones on RCD spaces with such inequalities are metric cones.

5 Non-smooth analysis and differentiable sphere theorems 1 Kei KONDO 2 Studying whether a topological sphere theorem can be a differentiable one, or not, we always face a difficulty, which is the fusion of exotic structure and cut loci: By Smale s h-cobordism theorem, any homotopoy sphere of dimension n 5 is a twisted sphere. So is any exotic sphere Σ n of dimension n > 4. By Weinstein s deformation technique for metrics, we see that any twisted sphere of general dimension admits a metric such that there is a point whose cut locus is one point. Since every Σ n (n > 4) also admits such a metric, we hardly notice the difference between Σ n and the standard n-sphere S n. For example, we can infer that a certain single cut point on the Grove-Shiohama type n- sphere 3 X is a big obstacle as a singular one whenever approximating a homeomorphism f between X and S n from the fact that X is twisted. Note that the f is, in fact, bi-lipschitz. The most important issue that we should address is therefore to do an analysis of such singular points on an arbitrary manifold. For that, we employ a notion used in non-smooth analysis established by F.H. Clarke, i.e., non-singular points for Lipschitz maps. The reason why we employ this notion is that it is a strong tool even in differential geometry. Let us give you an example of that: Let M be a complete Riemannian manifold, d its distance function, fix a p M. Put d p (x) := d(p, x) for all x M. Assume that, for some r > 0, the set (d p ) 1 (r) has no critical points of d p in the sense of Grove-Shiohama. By Gromov s isotopy lemma, (d p ) 1 (r) is a topological submanifold of M. Furthermore, by Clarke s implicit function theorem, we see that (d p ) 1 (r) is a Lipschitz manifold, for (d p ) 1 (r) is also free of critical points of d p in the sense of Clarke. In this talk, we prove, as a main theorem, that a Lipschitz map from a compact Riemannian manifold M into a Riemannian manifold N admits a smooth approximation via immersions if the map has no singular points on M in the Clarke sense, where dim M dim N. As its corollary, we have that if a bi-lipschitz homeomorphism between compact manifolds and its inverse have no singular points in the same sense, respectively, then they are diffeomorphic. As an indirect corollary of the theorem, we can construct a concrete bi-lipschitz homeomorphism F between any twisted n-sphere of general dimension and S n which is a diffeomorphism except for a single point. Therefore, the existence of Σ n (n > 4) implies that we cannot approximate F by diffeomorphisms. Hence, we must give careful consideration to sufficient conditions for a pair of topological spheres admitting a single cut point of some point, hence even for an exotic pair, to be diffeomorphic. As applications of the main theorem, we have two differentiable sphere theorems for such a pair including exotic pairs, where we will give the sufficient conditions for each pair to be diffeomorphic. Moreover, we have that if a compact n-manifold M satisfies certain two conditions for critical points of its distance function in the Clarke sense, then M is a twisted sphere and there exists a bi-lipschitz homeomorphism between M and the unit n-sphere S n (1) R n+1 which is a diffeomorphism except for a single point. As corollaries from this result, we observe that if an exotic 4-sphere Σ 4 exists, then Σ 4 does not satisfy one of the two conditions in the result; furthermore we do that for any Grove-Shiohama type n-sphere X, there exists a bi-lipschitz homeomorphism between X and S n (1) which is a diffeomorphism except for one of points that attain their diameters. This is joint work, arxiv: , with Minoru Tanaka. 1 At RIMS, Project conference Geometric analysis on Riemannian and metric spaces, 5 9 Sep., Department of mathematical sciences, Yamaguchi university, keikondo@yamaguchi-u.ac.jp 3 I.e., X is a complete Riemannian manifold with sectional curvature 1 and its diameter > π/2.

6 Monotonicity and Rigidity of the W-entropy on RCD (0, N) spaces Kazumasa Kuwada (School of Science, Tokyo Institute of Technology) Perelman s W-entropy plays a crucial role in his seminal work on Ricci flow. It is well-known by Perelman s entropy formula that the W-entropy is non-increasing in time and a time derivative vanishes if and only if the space is isomorphic to a gradient shrinking Ricci soliton. L. Ni brought the notion of W-entropy to time-homogeneous Riemannian manifolds, and the corresponding results has been studied in the literature under nonnegative Ricci curvature in an appropriate sense. In this talk, we consider the corresponding problem on RCD (0, N) metric measure spaces. RCD (0, N) space is a class of infinitesimally Hilbertian metric measure spaces with the nonnegative Ricci curvature and the upper bound of dimension by N, defined in terms of optimal transport. It includes all (weighted) Riemannian manifold (or smooth metric measure spaces) with nonnegative N-Bakry-Émery Ricci tensor, and Ricci limit spaces with an appropriate curvature-dimension bound. By following Topping s approach to this problem by optimal transport, we prove the monotonicity of the W-entropy without deriving the entropy formula. Moreover, we also show a rigidity of this monotonicity. Unlike the smooth case, some other singular spaces than Euclidean spaces admit a vanishing time derivative of the W-entropy. Our result is new even on a weighted Riemannian manifold in the sense that we require no additional bounded geometry assumption which is used to derive the entropy formula. This is a joint work with Xiang-Dong Li (Chinese Academy of Science). URL: kuwada@math.titech.ac.jp

7 Xiang-Dong Li (Chinese Academy of Sciences) W -entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds Inspired by Perelman s seminal work on the entropy formula for the Ricci flow, we prove the W -entropy formula for the heat equation of the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W -entropy formula for the heat equation of the time dependent Witten Laplacian on compact manifolds equipped with the (K, m)-super Ricci flows, where K R and m [n, ]. Furthermore, we prove an analogue of the W -entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W -entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the W -entropy formula for the Langevin deformation. Rigidity models are also provided. Joint work with Songzi Li. Ayato Mitsuishi (Gakushuin) Orientabilities and fundamental classes of Alexandrov spaces and its applications A study of Alexandrov spaces is important in Riemannian geometry. Here, an Alexandrov space is an intrinsic metric space having a lower curvature bound in a synthetic sense. Collapsing complete Riemannian manifolds with a uniform lower sectional curvature bound and the quotient spaces of complete Riemannian manifolds by isometric actions are typical Alexandrov spaces. I will exhibit several valid notions of orientability on Alexandrov spaces, which are global topological properties. Further, I will announce that those properties are equivalent to each other, and give topological and geometric applications to the orientability. Andrea Mondino (ETH Zurich) On the fundamental group of non smooth spaces with Ricci curvature bounded from below After reviewing some basics about the so-called RCD (K, N)-spaces, I will discuss some recent developments (joint work with G. Wei) about understanding the topology of such non-smooth spaces. The main theorem is the existence of the universal cover, which implies several structure results on the (revised) fundamental group of such spaces. Ovidiu Munteanu (Connecticut) Ricci Solitons This is a survey of recent progress on the geometry of shrinking Ricci solitons, with more emphasis on the development in dimension four. It is hoped that this study will help understand the possible singularities of the Ricci flow on four dimensional manifolds.

8 Shin-ichi Ohta (Kyoto) Nonlinear geometric analysis on Finsler manifolds: Gaussian isoperimetric inequality A Finsler manifold is a manifold equipped with a (Minkowski) norm on each tangent space. Although the natural Laplacian is nonlinear for Finsler manifolds, one can establish the Bochner formula (O.-Sturm, 2014) and use it to develop the nonlinear analogue of the Γ-calculus (of Bakry et al). In this talk, we mainly explain a geometric application to Bakry-Ledoux s Gaussian isoperimetric inequality, being sharp even for non-reversible metrics. Xiaochun Rong (CNU and Rutgers) Quantitative volume space form rigidity with Ricci curvature bounded below A complete Riemannian manifold of constant sectional curvature is referred as a space form. Space forms have been playing a fundamental role in Riemannian geometry. In this talk, we will formulate a new volume space form rigidity with lower Ricci curvature bound, and report some recent work on the quantitative version. This work is joint with Lina Chen and Shicheng Xu of Capital Normal University (ref. arxiv.org/abs/ ). Karl-Theodor Sturm (Bonn) Heat flow on time-dependent mm-spaces and super Ricci flows We study the heat equation on time-dependent metric measure spaces (being a dynamic forward gradient flow for the energy) and its dual (being a dynamic backward gradient flow for the Boltzmann entropy). Monotonicity estimates for transportation distances and for gradients will be shown to be equivalent to the so-called dynamical convexity of the Boltzmann entropy on the Wasserstein space. For time-dependent families of Riemannian manifolds the latter is equivalent to be a super-ricci flow i.e. to satisfy 2Ric + t g 0. This includes all static manifolds of nonnegative Ricci curvature as well as all solutions to the Ricci flow equation. We extend this concept of super-ricci flows to time-dependent metric measure spaces and prove stability and compactness of super-ricci flows under mgh-limits.

9 Some Myers Type Theorems and Hitchin Thorpe Inequalities for Shrinking Ricci Solitons Homare TADANO Department of Mathematics, Osaka University Geometric Analysis on Riemannian and Metric Spaces Abstract Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models. The importance of Ricci solitons was demonstrated by G. Perelman, where Ricci solitons played crucial roles in his affirmative resolution of the Poincaré conjecture. Besides their geometric importance, Ricci solitons are also of great interest in theoretical physics and have been studied actively in relation to string theory. In this talk, after we reviewed basic facts on Ricci solitons, we shall establish some compactness theorems for complete shrinking Ricci solitons [9, 10, 11]. Our compactness theorems generalize previous ones obtained by Fernández- López and García-Río [2], Wei and Wylie [13], Limoncu [4, 5], Rimoldi [6] and Zhang [14]. As applications of these compactness theorems, we shall give some upper diameter bounds for compact Ricci solitons. Moreover, by using such diameter bounds, we shall provide some new sufficient conditions for four-dimensional compact Ricci solitons to satisfy the Hitchin Thorpe inequality. If time permits, we shall consider some natural generalizations of Ricci solitons such as quasi Einstein manifolds [1] and Sasaki Ricci solitons [3] and we shall give a gap theorem and a diameter bound for such manifolds [7, 8, 12]. References [1] J. Case, Y.-J. Shu and G. Wei, Rigidity of quasi Einstein metrics, Diff. Geom. Appl. 29 (2011) [2] M. Fernández López and E. García Río, A remark on compact Ricci solitons, Math. Ann. 340 (2008) [3] A. Futaki, H. Ono and G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki Einstein manifolds, J. Differential Geom. 83 (2009) [4] M. Limoncu, Modifications of the Ricci tensor and applications, Arch. Math. (Basel) 95 (2010) [5], The Bakry Emery Ricci tensor and its applications to some compactness theorems, Math. Z. 271 (2012) [6] M. Rimoldi, A remark on Einstein warped products, Pacific J. Math. 252 (2011) [7] H. Tadano, Gap theorems for compact gradient Sasaki Ricci solitons, Internat. J Math. 26 (2015) [8], Diameter bounds, gap theorems and Hitchin Thorpe inequalities for compact quasi Einstein manifolds, Preprint (2015) [9], Remark on a diameter bound for complete Riemannian manifolds with positive Bakry Emery Ricci curvature, Diff. Geom. Appl. 44 (2016) [10], An upper diameter bound for compact Ricci solitons with applications to the Hitchin Thorpe inequality, Preprint (2016) [11], Some Ambrose and Galloway type theorems via Bakry Émery and modified Ricci curvatures, Preprint (2016) [12], Lower diameter bounds and transverse Hitchin Thorpe inequalities for compact gradient Sasaki Ricci solitons, in preparation (2016) [13] G. Wei and W. Wylie, Comparison geometry for the Bakry Emery Ricci tensor, J. Differential Geom. 83 (2009) [14] S. Zhang, A theorem of Ambrose for Bakry Emery Ricci tensor, Ann. Global Anal. Geom. 45 (2014)

10 William Wylie (Syracuse) Weighted Ricci curvature with effective dimension one on smooth manifolds We study smooth manifolds with density that satisfy weighted Ricci curvature bounds of effective dimension one, a weaker condition than has previously been studied. The main observation is that the weighted Ricci tensor in this case carries an extra structure, as it is the Ricci tensor of a natural torsion free connection. Using this connection we are not only able to prove generalizations of basic comparison results such as the splitting theorem, Myers Theorem, the maximal diameter theorem, and Bishop-Gromov volume comparison theorem, but we also obtain new structures to study, such as the holonomy group of connection. We also study the basic properties of the holonomy groups and prove a version of the de Rham splitting theorem. A general feature of all of our rigidity results is that warped or twisted products, as opposed to direct products are characterized. This is joint work with Dmytro Yeroshkin, a former post-doc at Syracuse who is now at Idaho State University. Takao Yamaguchi (Kyoto) Inradius collapsed manifolds In this talk, we will consider collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and an upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds with boundary. First, we determine the limit spaces of inradius collapsed manifolds. Secondly, when the limit space has co-dimension one, we determine the topology of inradius collapsed manifold. Thirdly, we prove a boundary rigidity result in the unbounded diameter case. This is a joint work with Zhilang Zhang.