标题： 超定热方程的刚性及空间形式中的极小螺旋面 显示英文标题

标题： Rigidity of an overdetermined heat equation and minimal helicoids in space-forms

摘要： 设$M$为一个黎曼流形，$\Omega$为$M$的一个光滑区域。 我们研究以下热扩散问题：假设初始温度在$\Omega$上均匀等于$1$，并在其补集上为$0$。 热量将从$\Omega$流向其补集，我们感兴趣的是$\Omega$边界在所有正时间$t>0$的温度。 特别是我们问：是否存在这样的区域，使得边界上的温度在所有正时间$t$和边界上的所有点都是常数$c$？ 如果存在，我们可以对它们的几何形状说些什么？ 这是一个典型的超定热方程的例子。 显然，如果$c$存在，它必须是$\frac 12$，具有恒定边界温度的区域将被称为具有$\frac 12$-性质。 以前的工作由\cite{MPS06}和\cite{CSU23}表明，在$\mathbb R^3$上，唯一这样的区域（至合同）的边界是一平面或（有点出人意料的是）右螺旋面。 在本文中，我们首先证明，在很大程度上，$\frac 12$-域的边界必须是极小的；然后我们将上述分类从$\mathbb R^3$扩展到其他$3$-维空间形式（并给出了不同的证明）。 我们证明，在$\mathbb S^3$中，$\frac 12$-域由全测地曲面或克莱弗兰环面限定，在双曲空间$\mathbb H^3$中，它们由全测地曲面或一个（嵌入的）双曲螺旋面限定。 %(存在这样曲面的一个参数族) 作为副产品，我们扩展(用不同的证明)了Nitsche关于均匀稠密区域的结果，从$\mathbb R^3$到$3$维空间形式。 显示英文摘要

摘要： Let $M$ be a Riemannian manifold and $\Omega$ a smooth domain of $M$. We study the following heat diffusion problem: assume that the initial temperature is equal to $1$, uniformly on $\Omega$, and is $0$ on its complement. Heat will then flow away from $\Omega$ to its complement, and we are interested in the temperature on the boundary of $\Omega$ at all positive times $t>0$. In particular we ask: are there domains for which the temperature at the boundary is a constant $c$, for all positive times $t$ and for all points of the boundary? If they exist, what can we say about their geometry? This is a typical example of overdetermined heat equation. It is readily seen that if $c$ exists it must be $\frac 12$, and domains with constant boundary temperature will be said to have the $\frac 12$-property. Previous work by \cite{MPS06} and \cite{CSU23} show that, on $\mathbb R^3$, the only such domains (up to congruences) have boundary which is a plane or (a bit surprisingly) the right helicoid. In this paper we first show that, in great generality, the boundary of a $\frac 12$-domain must be minimal; we then extend (with a different proof) the above classification from $\mathbb R^3$ to the other $3$-dimensional space-forms. We prove that, in $\mathbb S^3$, $\frac 12$-domains are bounded by a totally geodesic surface or the Clifford torus, and in the hyperbolic space $\mathbb H^3$ are bounded by a totally geodesic surface or by an (embedded) minimal hyperbolic helicoid. %(there is a one-parameter family of such surfaces) As a by-product, we extend (with a different proof) a result by Nitsche on uniformly dense domains from $\mathbb R^3$ to $3$-dimensional space-forms.