A Sharp Li-Yau gradient bound on Compact Manifolds
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| Format: | Preprint |
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2021
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| _version_ | 1866909442456420352 |
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| author | Zhang, Qi S. |
| author_facet | Zhang, Qi S. |
| contents | Let $(\M^n, g)$ be a $n$ dimensional, complete ( compact or noncompact) Riemannian manifold whose Ricci curvature is bounded from below by a constant $-K \le 0$. Let $u$ be a positive solution of the heat equation on $\M^n \times (0, \infty)$. The well known Li-Yau gradient bound states that $$ t \left(\frac{|\nabla u|^2}{u^2} - α\frac{\pa_t u}{u}\right) \leq \frac{nα^2}{2} + t \frac{nα^2K}{2(α-1)},\quad \forall α>1, t>0. $$ The bound with $α=1$ is sharp if $K=0$. If $-K < 0$, the bound tends to infinity if $α=1$. In over 30 years, several sharpening of the bounds have been obtained with $α$ replaced by several functions $α=α(t)>1$ but not equal to $1$. An open question (\cite{CLN}, \citeLX} etc) asks if a sharp bound can be reached. In this short note, we observe that for all complete compact manifolds one can take $α=1$. Thus a sharp bound, up to computable constants, is found in the compact case. This result also seems to sharpen Theorem 1.4 in \cite{LY} for compact manifolds with convex boundaries. In the noncompact case one can not take $α=1$ even for the hyperbolic space. An example is also given, which shows that there does not exist an optimal function of time only $α=α(t)$ for all noncompact manifolds with Ricci lower bound, giving a negative answer to the open question in the noncompact case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_08933 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Sharp Li-Yau gradient bound on Compact Manifolds Zhang, Qi S. Differential Geometry Analysis of PDEs 53C44 Let $(\M^n, g)$ be a $n$ dimensional, complete ( compact or noncompact) Riemannian manifold whose Ricci curvature is bounded from below by a constant $-K \le 0$. Let $u$ be a positive solution of the heat equation on $\M^n \times (0, \infty)$. The well known Li-Yau gradient bound states that $$ t \left(\frac{|\nabla u|^2}{u^2} - α\frac{\pa_t u}{u}\right) \leq \frac{nα^2}{2} + t \frac{nα^2K}{2(α-1)},\quad \forall α>1, t>0. $$ The bound with $α=1$ is sharp if $K=0$. If $-K < 0$, the bound tends to infinity if $α=1$. In over 30 years, several sharpening of the bounds have been obtained with $α$ replaced by several functions $α=α(t)>1$ but not equal to $1$. An open question (\cite{CLN}, \citeLX} etc) asks if a sharp bound can be reached. In this short note, we observe that for all complete compact manifolds one can take $α=1$. Thus a sharp bound, up to computable constants, is found in the compact case. This result also seems to sharpen Theorem 1.4 in \cite{LY} for compact manifolds with convex boundaries. In the noncompact case one can not take $α=1$ even for the hyperbolic space. An example is also given, which shows that there does not exist an optimal function of time only $α=α(t)$ for all noncompact manifolds with Ricci lower bound, giving a negative answer to the open question in the noncompact case. |
| title | A Sharp Li-Yau gradient bound on Compact Manifolds |
| topic | Differential Geometry Analysis of PDEs 53C44 |
| url | https://arxiv.org/abs/2110.08933 |