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We establish the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with $\epsilon$-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie-Yeroshkin and Kuwae-Li., Comment: 39 pages; minor revisions, to appear in Anal. Geom. Metr. Spaces

We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz-Finsler version of the Bonnet-Myers theorem based on a generalized Bishop inequality., Comment: 37 pages; some modifications to clarify motivation and improve presentation; to appear in J. Lond. Math. Soc

Concerning quantitative isoperimetry for a weighted Riemannian manifold satisfying $\mathrm{Ric}_{\infty} \ge 1$, we give an $L^1$-estimate exhibiting that the push-forward of the reference measure by the guiding function (arising from the needle decomposition) is close to the Gaussian measure. We also show $L^p$- and $W_2$-estimates in the $1$-dimensional case., v2: 10 pages, to appear in Bull. Lond. Math. Soc

This chapter is devoted to a geometric application of the improved Bochner inequality, the Bakry–Ledoux isoperimetric inequality (also called the Gaussian isoperimetric inequality). This is one of the most important geometric applications of the Γ-calculus. The asymptotic behavior of (nonlinear or linearized) heat semigroups for large time will play an essential role. A related analysis also shows the Poincare–Lichnerowicz inequality.

Comparison theorems are the main subjects of this book. They are concerned with quantitative or qualitative properties of a space with a certain condition on its curvature. In this book, we will consider Finsler manifolds whose flag or (weighted) Ricci curvature is bounded from below or above by a constant. This chapter is devoted to some fundamental examples of geometric comparison theorems. The first two of them (the Bonnet–Myers and Cartan–Hadamard theorems) are verbatim analogues of the Riemannian counterparts. Then we study the convexity and concavity of the distance function using some non-Riemannian quantities besides the flag curvature.

Since the end of twentieth century, optimal transport theory has been making a breathtaking and diverge progress in and outside mathematics, e.g., partial differential equations, probability theory, differential geometry, economics, and image processing, to name a few. What is especially relevant to our interest is the convexity of an entropy functional along optimal transports, called the curvature-dimension condition. This notion, due to Lott, Sturm, and Villani, turned out having rich applications in analysis and geometry. In this chapter, in our framework of Finsler manifolds, we overview the basic ideas of optimal transport theory and its relation with the weighted Ricci curvature.

In this chapter, we discuss fundamental properties of the nonlinear heat equation ∂tu = Δu associated with the nonlinear Laplacian Δ defined in Chap. 11. In particular, we establish the existence and the regularity of global solutions to the heat equation. Coupled with the Bochner inequalities in the previous chapter, the analysis of heat flow leads to various analytic and geometric applications as we will see in the following chapters. We remark that, due to the nonlinearity, there is no heat kernel.

This chapter is devoted to some fundamental and important examples of Finsler manifolds. In addition to Minkowski normed spaces and Randers spaces, we introduce Berwald spaces, Hilbert and Funk geometries, and Teichmuller spaces and discuss their characteristic properties.

In this chapter, we analyze the weighted Ricci curvature for some examples appearing in Chap. 6. We remark that a suitable choice of a measure is unclear in some cases. In fact, the theory of weighted Ricci curvature has so far been focused on general theory and there are not many investigations on concrete examples.

In this chapter, we consider the natural energy functional (for functions) and the corresponding Sobolev spaces. Then we introduce the nonlinear Laplacian in a way that its associated harmonic functions are minimizers of the energy functional. We also show the Laplacian comparison theorem as the first analytic comparison theorem.

In this chapter, as a warm-up before the general theory of Finsler manifolds, we consider normed spaces and discuss some characterizations of inner product spaces among normed spaces. These special properties of inner product spaces will help us to understand the difference between Riemannian and Finsler manifolds.

In this chapter we study the first and second variation formulas for arclength, followed by some applications to the behavior of the distance function along geodesics, including the study of cut and conjugate points. The first variation formula is closely related to the geodesic equation, which was introduced as the Euler–Lagrange equation for the energy functional. The second variation formula will be related to the flag curvature.

This chapter is devoted to the main ingredients of our geometric analysis on measured Finsler manifolds, the Bochner–Weitzenbock formula (or the Bochner formula), and the corresponding Bochner inequality, in terms of the nonlinear Laplacian and its linearization introduced in the previous chapter. In the language of the celebrated Γ-calculus a la Bakry et al., the Bochner inequality can be regarded as a nonlinear analogue of the Γ2-criterion.

In this chapter, we begin our study of differential calculus on Finsler manifolds. The main subject of the chapter is the geodesic equation as the Euler–Lagrange equation for the energy functional. To this end, some important quantities such as the fundamental and Cartan tensors are introduced. We will see that the metric definition of geodesics coincides with the variational definition as solutions to the geodesic equation. We also prove the Finsler analogue of the Hopf–Rinow theorem.

We investigate self-contracted curves, arising as (discrete or continuous-time) gradient curves of quasi-convex functions, and their rectifiability (finiteness of the lengths) in Euclidean spaces, Hadamard manifolds, and $${\mathrm {CAT}}(0)$$-spaces. In the Hadamard case, we give a quantitative refinement of the original proof of the rectifiability of bounded self-contracted curves (in general Riemannian manifolds) by Daniilidis et al. Our argument leads us to a generalization to $${\mathrm {CAT}}(0)$$-spaces satisfying several uniform estimates on their local structures. Upon these conditions, we show the rectifiability of bounded self-contracted curves in trees, books, and $${\mathrm {CAT}}(0)$$-simplicial complexes.

This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is twofold: give a concise and geometric review on the birth of weighted Ricci curvature and its applications; explain recent results from a nonlinear analogue of the \({\Gamma }\)-calculus based on the Bochner inequality. In the latter we discuss some gradient estimates, functional inequalities, and isoperimetric inequalities.

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present several related open problems., 13pages; to appear in manuscripta mathematica

This is a survey article on recent progress (in [Oh3], [OS]) of the theory of weighted Ricci curvature in Finsler geometry. Optimal transport theory plays an impressive role as is developed in the Riemannian case by Lott, Sturm and Villani.

We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author's previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. The results in this article show that such a strategy applies to spaces with lower curvature bound as well., Comment: A goofy mistake in formulations of Theorem 2 and 4 is fixed

We prove the Bochner–Weitzenbock formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li–Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry–Emery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent \(n \ge 3\), then it has exactly the \(n\)-dimensional volume growth. As an application, if an \(n\)-dimensional Finsler manifold of non-negative \(n\)-Ricci curvature satisfies the Caffarelli–Kohn–Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.

We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces., 21 pages; minor corrections; to appear in J. Reine Angew. Math

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the $\Gamma$-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti--Mondino in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics., Comment: 22 pages (no change from v3); Corrigendum is available at http://www4.math.sci.osaka-u.ac.jp/~sohta/ ; See also my book "Comparison Finsler geometry" https://link.springer.com/book/10.1007/978-3-030-80650-7

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication., Comment: 7 pages (no change from v1); Corrigendum is available at http://www4.math.sci.osaka-u.ac.jp/~sohta/ ; See also my book "Comparison Finsler geometry" https://link.springer.com/book/10.1007/978-3-030-80650-7

eight conformational isomersSynthesis of a deuterium-labeled derivative of nitrospirobenzopyran (NSP), one of representative photochromic compounds, has been described. Four deuteriums were successfully introduced on 1-methyl and α -methyne relative to spiro-carbon in the title compound with more than 95atom%D purity. Main photodegraded products of NSP were two oxindoles in acetonitrile, and additional products were formed in poly(isobutyl-methacrylate) films possibly due to restricted molecular motion in polymer matrix. Quantitative HPLC analysis revealed that partial introduction of deuterium to NSP brought a noticeable isotope effect, recognizable enhancement in photo-resistivity of NSP, i.e., 8.3% in solutions and 29% in polymeric films. Latent durability of synthetic compositions has become a big issue to utilize them in state-of-the-art, optoelectronic devices for stable long-term usage so that technological development toward robustness of molecular based materials is quite important. We focus attention on isotopic derivatization of organic molecules through recently developed catalytic protium(H)-deuterium(D) exchange reactions in D

We prove that Alexandrov spaces of non-negative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of an Alexandrov space of non-negative curvature into a 2-uniformly convex Banach space can be extended to a Lipschitz continuous map on the entire space.

We extend Cordero-Erausquin et al.'s Riemannian Borell-Brascamp-Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani's curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.

We establish the existence of Euclidean tangent cones on Wasserstein spaces over compact Alexandrov spaces of curvature bounded below. By using this Riemannian structure, we formulate and construct gradient flows of functions on such spaces. If the underlying space is a Riemannian manifold of nonnegative sectional curvature, then our gradient flow of the free energy produces a solution of the linear Fokker-Planck equation.

We prove that, if a geodesic metric space has Markov type 2 with constant 1, then it is an Alexandrov space of nonnegative curvature. The same technique provides a lower bound of the Markov type 2 constant of a space containing a tripod or a branching point.

We generalize the Alexandrov–Toponogov comparison theorems to Finsler manifolds. Under suitable upper (lower, resp.) bounds on the flag and tangent curvatures together with the 2-uniform convexity (smoothness, resp.) of tangent spaces, we show the 2-uniform convexity (smoothness, resp.) of Finsler manifolds. As applications, we prove the almost everywhere existence of the second order differentials of semi-convex functions and of c-concave functions with the quadratic cost function.

We introduce a stochastic generalization of Lipschitz retracts, and apply it to the extension problems of Lipschitz, Holder, large-scale Lipschitz and large-scale Holder maps into barycentric metric spaces. Our discussion gives an appropriate interpretation of a work of Lee and Naor.

We investigate the topology of a complete Riemannian manifold whose radial curvature at the base point is bounded from below by that of a von Mangoldt surface of revolution. Sphere theorem is generalized to a wide class of metrics, and it is proven that such a manifold of a noncompact type has finitely many ends.

We introduce two kinds of the notion of convexity of a metric space, called k-convexity and L-convexity, as generalizations of the CAT(0)-property and of the nonpositively curved property in the sense of Busemann, respectively. 2-uniformly convex Banach spaces as well as CAT(1)-spaces with small diameters satisfy both these convexities. Among several geometric and analytic results, we prove the solvability of the Dirichlet problem for maps into a wide class of metric spaces.

Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by optimal transport theory. Cavalletti and Mondino subsequently generalized the localization method, in a different way more directly along optimal transport theory, to essentially non-branching metric measure spaces satisfying the curvature-dimension condition. This class in particular includes reversible (absolutely homogeneous) Finsler manifolds. In this paper, we construct needle decompositions of non-reversible (only positively homogeneous) Finsler manifolds, and show an isoperimetric inequality under bounded reversibility constants. A discussion on the curvature-dimension condition CD$(K,N)$ for $N = 0$ is also included, it would be of independent interest., 44 pages: minor revisions, to appear in J. Math. Soc. Japan

Organic light emitting diodes (OLEDs) with nanostructured ultrathin layers inserted at the interface between electron- and hole-transport layers were investigated. The fundamental structure of the OLEDs fabricated by a vacuum evaporation method was indium-tin-oxide (ITO) anode/copper phthalocyanine (CuPc)/N,N′-diphenyl-N,N′-bis(3-methylphenyl)-1,1′-diphenyl-4,4′-diamine (TPD)/8-hydroxyquinoline aluminum (Alq3)/LiF/Al cathode. Fullerene (C60) and rhodamine B (RhB) molecules were used as the nanosutructured ultrathin layers inserted at the interface between the Alq3 and TPD layers. The electroluminescent (EL) properties have been measured for the OLEDs with C60 and RhB ultrathin layers and the dependences on the thickness and the position of the inserted layers were examined. For the OLEDs with the C60 ultrathin layer, the improvements of the drive voltage and EL efficiency were observed. The OLED with the inserted C60 ultrathin film of a monolayer thickness showed the highest efficiency, which was twice as large as that without C60 layer. On the contrary, the improvements were not observed for the OLEDs with the RhB ultrathin layer.

We give a geometric approach to the study of the regularity of harmonic functions in Cheeger-type Sobolev spaces, and prove the Holder continuity of such functions. In the proof, we give a definition of an upper curvature bound of the unit sphere of a Banach space, which seems to be of independent interest.

Many amorphous molecular systems prepared by vacuum deposition onto cold substrates turn into crystals directly without exhibiting glass transition when their temperature is raised. To study the me...