A new Laplacian comparison theorem and its applications on Finsler manifold (Q6569661)

In this paper the authors investigate a Finsler \(n\)-manifold \((M, F, d\mu)\) with an arbitrary volume form \(d\mu = \sigma(x) dx\). Let \(p\) be a fixed point and \(r(x) = d(p, x)\) the distance function from \(p\). The authors assume that \(d\tilde{\mu} = \tilde{\sigma}(x)\, dx\), where \(\tilde{\sigma}(x) = \sigma(x) e^{\tau(x, \nabla r(x))}\), and \(\tau\) is the distortion with respect to \(d\mu\). Then \(d\tilde{\mu}\) gives a weighted volume form on \(M \setminus \{p\}\). Since, for any integrable function \(f\), we have \( \int_{M \setminus \{p\}} f\, d\tilde{\mu} = \int_M f\, d\tilde{\mu} \), we can also consider \(\tilde{\mu}\) as a volume form defined on \(M\)\N\NThe main aim of this article is to establish a new Laplacian comparison theorem on Finsler manifolds with weighted volume form \(\tilde{\mu}\). Precisely, we have\N\NTheorem 1.1: Let \((M, F, d\mu)\) be a Finsler \(n\)-manifold with an arbitrary volume form \(d\mu = \sigma(x) \,dx\). Let \(p\) be a fixed point and \(r(x) = d(p, x)\) the distance function from \(p\). Assume that the volume form \(\tilde{\mu} = \tilde{\sigma}(x) \,dx\), where \(\tilde{\sigma}(x) = \sigma(x)e^{\tau(x, \nabla r(x))}\), and \(\tau\) is the distortion with respect to \(d\mu\). If the Ricci curvature satisfies \(\mathrm{Ric} \geq (n-1)k\), then the Laplacian of \(r(x)\) with respect to \(\tilde{\mu}\) can be estimated as follows: \N\[\N\tilde{\Delta} r \leq (n-1)\mathrm{ct}_k(r)\N\]\Npointwise on \(M \setminus (\{p\} \cup \mathrm{Cut}(p))\) and in the sense of distributions on \(M \setminus \{p\}\). Here \N\[\N\mathrm{ct}_k(r) = \begin{cases} \sqrt{k} \cot(\sqrt{k}r), & k > 0, \\\N\frac{1}{r}, & k = 0, \\\N\sqrt{-k} \coth(\sqrt{-k}r), & k < 0. \end{cases}\N\]\NThe equality holds if and only if the radial flag curvature satisfies \(K(x, \nabla r(x)) = k\).\N\NThe second aim of this paper is to give some applications of the Laplacian comparison theorem (Theorem 1.1). In particular, the authors obtain a volume comparison theorem as follows.\N\NTheorem 1.2: Let \((M, F, d\mu)\) be a Finsler \(n\)-manifold with an arbitrary volume form \(d\mu = \sigma(x) dx\). Let \(p\) be a fixed point and \(r(x) = d(p, x)\) the distance function from \(p\). Assume that the volume form \(\tilde{\mu} = \tilde{\sigma}(x) dx\), where \(\tilde{\sigma}(x) = \sigma(x)e^{\tau(x, \nabla r(x))}\), and \(\tau\) is the distortion with respect to \(d\mu\). If the Ricci curvature satisfies \(\mathrm{Ric} \geq (n-1)k\), then for any \(0 < r < R \left(R \leq \frac{\pi}{\sqrt{k}} \text{ when } k > 0\right)\), it holds that\N\[\N\frac{\mathrm{vol}_{F}^{\tilde{\mu}} B_p^+(R)}{\mathrm{vol}_{F}^{\tilde{\mu}} B_p^+(r)} \leq \frac{\int_0^R s_k(t)^{n-1} dt}{\int_0^r s_k(t)^{n-1} dt},\N\]\Nwhere \(B_p^+(r)\) denotes the forward geodesic ball centered at \(p\) of radius \(r\), and \(s_k(t)\) is defined by (3.1) below. The equality holds if and only if the radial flag curvature satisfies \(K(x, \nabla r(x)) = k\).