Heat flow on Finsler manifolds

@article{Ohta2008HeatFO,
  title={Heat flow on Finsler manifolds},
  author={Shin-Ichi Ohta and Karl-Theodor Sturm},
  journal={Communications on Pure and Applied Mathematics},
  year={2008},
  volume={62}
}
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… 

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References

SHOWING 1-10 OF 42 REFERENCES

The Non-Linear Laplacian for Finsler Manifolds

For a Finsler manifold (M,F), there is a canonical energy function E defined on the Sobolev space. The variation of E gives rises to a non-linear Laplacian. Although this Laplacian is non-linear, it

Ricci curvature for metric-measure spaces via optimal transport

We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the

Uniformly elliptic operators on Riemannian manifolds

Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g .

An Introduction to Riemann-Finsler Geometry

One Finsler Manifolds and Their Curvature.- 1 Finsler Manifolds and the Fundamentals of Minkowski Norms.- 1.0 Physical Motivations.- 1.1 Finsler Structures: Definitions and Conventions.- 1.2 Two

On the geometry of metric measure spaces. II

AbstractWe introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m}

Weak curvature conditions and functional inequalities

Uniform convexity and smoothness, and their applications in Finsler geometry

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

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence

Gradient flows on Wasserstein spaces over compact Alexandrov spaces

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