• DOI:10.1002/CPA.20273
• Corpus ID: 14124587

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}
}

• Shin-Ichi Ohta, K. Sturm
• Published 8 August 2008
• Mathematics
• Communications on Pure and Applied Mathematics

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.