Report of Workshop on Finsler Geometry in Kyoto, 2011 (in Japanese)

Differential Topology Seminar (in Japanese only)

**Keywords:**- Riemannian geometry, Alexandrov spaces, CAT-spaces,
analysis on metric measure spaces,

optimal transport theory, Wasserstein geometry, gradient flows,

Finsler geometry, the geometry of Banach spaces, ...

- Needle decompositions and isoperimetric inequalities
in Finlser geometry, June 2015

(Revised at 7 Jul 2015: minor revisions in abstract & introduction; PDF). - (with Miklós PÁLFIA)
Gradient flows and a Trotter-Kato formula
of semi-convex functions on CAT(1)-spaces, Oct 2014

(Revised at 7 Nov 2014: references added; PDF). - (with Miklós PÁLFIA)
Discrete-time gradient flows and law of large numbers
in Alexandrov spaces, Feb 2014
(Revised at 25 Feb 2015;
PDF),

to appear in*Calc. Var. Partial Differential Equations.* - (
*K,N*)-convexity and the curvature-dimension condition for negative*N*, Oct 2013 (Revised at 29 May 2015; PDF),

to appear in*J. Geom. Anal.* - (with Kei KONDO & Minoru TANAKA) A Toponogov type triangle comparison theorem in Finsler geometry, May 2012 (PDF).

- Splitting theorems for Finsler manifolds
of nonnegative Ricci curvature,
*J. Reine Angew. Math.***700**(2015), 155-174 (PDF). - (with Kei KONDO & Minoru TANAKA)
Topology of complete Finsler manifolds
with radial flag curvature bounded below,
*Kyushu J. Math.***68**(2014), 347-357 (PDF). - On the curvature and heat flow on Hamiltonian systems,
*Anal. Geom. Metr. Spaces***2**(2014), 81-114 (PDF). - Examples of spaces with branching geodesics
satisfying the curvature-dimension condition,
*Bull. Lond. Math. Soc.***46**(2014), 19-25 (PDF). - (with Karl-Theodor STURM)
Bochner-Weitzenböck formula and Li-Yau estimates
on Finsler manifolds,
*Adv. Math.***252**(2014), 429-448 (PDF). - (with Alexandru KRISTÁLY)
Caffarelli-Kohn-Nirenberg inequality
on metric measure spaces with applications,
*Math. Ann.***357**(2013), 711-726 (PDF). - Weighted Ricci curvature estimates for Hilbert and Funk geometries,
*Pacific J. Math.***265**(2013), 185-197 (PDF). - (with Asuka TAKATSU)
Displacement convexity of generalized relative entropies. II,
*Comm. Anal. Geom.***21**(2013), 687-785 (PDF). - (with Nicola GIGLI & Kazumasa KUWADA)
Heat flow on Alexandrov spaces,
*Comm. Pure Appl. Math.***66**(2013), 307-331 (PDF). - (with Karl-Theodor STURM)
Non-contraction of heat flow on Minkowski spaces,
*Arch. Ration. Mech. Anal.***204**(2012), 917-944 (PDF). - (with Nicola GIGLI)
First variation formula in Wasserstein spaces over compact
Alexandrov spaces,
*Canad. Math. Bull.***55**(2012), 723-735 (PDF). - Barycenters in Alexandrov spaces of curvature bounded below,
*Adv. Geom.***12**(2012), 571-587 (PDF). - (with Asuka TAKATSU)
Displacement convexity of generalized relative entropies,
*Adv. Math.***228**(2011), 1742-1787 (PDF). - Vanishing S-curvature of Randers spaces,
*Differential Geom. Appl.***29**(2011), 174-178 (PDF). - (with Karl-Theodor STURM) Heat flow on Finsler manifolds,
*Comm. Pure Appl. Math.***62**(2009), 1386-1433

(PDF; minor revisions in Lemma 3.8 & Theorem 3.9). - Finsler interpolation inequalities,
*Calc. Var. Partial Differential Equations***36**(2009), 211-249 (PDF). - (with Mikaël PICHOT) A note on Markov type constants,
*Arch. Math. (Basel)***92**(2009), 80-88 (PDF). - Uniform convexity and smoothness, and their applications
in Finsler geometry,
*Math. Ann.***343**(2009), 669-699 (PDF). - Extending Lipschitz and Hölder maps between metric spaces,
*Positivity***13**(2009), 407-425 (PDF). - Gradient flows on Wasserstein spaces over compact Alexandrov
spaces,
*Amer. J. Math.***131**(2009), 475-516 (PDF). - Markov type of Alexandrov spaces of nonnegative curvature,
*Mathematika***55**(2009), 177-189 (PDF). - (with Kei KONDO) Topology of complete manifolds with radial
curvature bounded from below,
*Geom. Funct. Anal.***17**(2007), 1237-1247. - Products, cones, and suspensions of spaces
with the measure contraction property,
*J. Lond. Math. Soc. (2)***76**(2007), 225-236 (PDF). - On the measure contraction property of metric measure spaces,
*Comment. Math. Helv.***82**(2007), 805-828 (PDF). - Convexities of metric spaces,
*Geom. Dedicata***125**(2007), 225-250 (PDF). - Reconstructions of distances by energy forms,
*Proc. Amer. Math. Soc.***134**(2006), 3405-3415. - Regularity of harmonic functions in Cheeger-type Sobolev spaces,
*Ann. Global Anal. Geom.***26**(2004), 397-410. - Harmonicity of totally geodesic maps into nonpositively curved
metric spaces,
*Manuscripta Math.***114**(2004), 127-138. - Cheeger type Sobolev spaces for metric space targets,
*Potential Anal.***20**(2004), 149-175. - Totally geodesic maps into metric spaces,
*Math. Z.***244**(2003), 47-65.

- Harmonic maps and totally geodesic maps between metric spaces,
Dissertation, Tohoku University, 2003.

Published in*Tohoku Mathematical Publications***28**, Tohoku University, 2004. - Optimal transport and Ricci curvature in Finsler geometry,
*Adv. Stud. Pure Math.***57**(2010), 323-342 (PDF).

*--Survey article on "Finsler interpolation inequalities" and "Heat flow on Finsler manifolds".* - Ricci curvature, entropy and optimal transport
(Revised 28 Mar 2014;
PDF),

In:*London Mathematical Society Lecture Note Series***413**(2014), 145-199.

*--Extended lecture notes for Summer School in Grenoble 2009.*