Heat flow on Finsler manifolds

By Shin-ichi Ohta, Karl‐Theodor Sturm. 2008

222 citations recorded by the source corpus.

This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → ℝ+ on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: as gradient flow on L2(M, m) for the energy $$ \varepsilon(u) = {1 \over 2} \int \limits_{M} F^{2}(\nabla u)dm $$ as gradient flow on the reverse L2‐Wasserstein space 𝒫2(M) of probability measures on M for the relative entropy $$ {\rm Ent}(u) = \int \limits_{M} u \log u \,dm. $$ Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove 𝒞1, α regularity for solutions to the (nonlinear) heat equation on the Finsler space (M, F, m). Typically solutions to the heat equation will not be 𝒞2. Moreover, we derive pointwise comparison results à la Cheeger‐Yau and integrated upper Gaussian estimates à la Davies. © 2008 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/55b3d5630d1e0d07d932d12feb0998345941f643

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Citations in the corpus

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, background): “In connection with (ii), notice that the equivalence so far has been proved in Euclidean spaces by Jordan-Kinderleher-Otto, in the seminal paper [24], in Riemannian manifolds by Erbar and Villani [13, 41], in Hilbert spaces by [6], in Finsler spaces by Ohta-Sturm [32] and eventually in Alexandrov sp”
Heat Flow on Alexandrov Spaces (2010, background): “, it has been proved that in Rn, in Riemannian manifolds with Ricci curvature bounded below and in compact Finsler manifolds (for the latter see the work of Sturm and the third author [28]), the gradient flow of the Dirichlet energy with respect to the L2-distance coincides with the gradient flow of”
Optimal transport from Lebesgue to Poisson (2010, unknown): “[OV00], [AZ09], [OS09]) and geometry (e.”
Ricci Curvature on Alexandrov spaces and Rigidity Theorems (2009, methodology): “Furthermore, it was shown by Ohta in [22] that on a Finsler manifolds M , the curvature-dimension condition CD(n, k) is equivalent to the weighted Finsler Ricci curvature condition Ricn(M) k (see also [23] or [25], refer to [23] for the definition Ricn in Finsler manifolds).”
Finsler interpolation inequalities (2009, background): “This theorem would be even more significant than Theorem 1.1 from the geometric viewpoint and there are a lot of analytic and geometric applications (see [ 27 ] for recent progress), though we do not pursue that direction in this article.”
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds (2012, background): “The remarkable identification between (Ht)t≥0 and (Pt)t≥0 has been firstly proposed and proved in Euclidean spaces by a seminal paper of Jordan-Kinderleher-Otto [31] and then extended to Riemannian manifolds [23, 55], Hilbert spaces [7], Finsler spaces [44], Alexandrov spaces [28] and eventually to ”
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy (2011, background): “In this sense, W takes over the role of the Wasserstein metric, since it is known since the seminal work by Jordan, Kinderlehrer, and Otto that the heat flow on Rn is the gradient flow of the entropy [23] (see [2, 18, 19, 20, 32, 36] for variations and generalisations).”
Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces (2013, background): “As showed initially by [25] in R n and then extended to many different situations by [21, 36, 6, 30, 23], it turns out [4] that (Ht)t≥0 essentially coincides with the L2-gradient flow (Pt)t≥0 of the convex and lower semicontinuous Cheeger energy Ch(f) := inf {”