Rigidity of an overdetermined heat equation and minimal helicoids in space-forms
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2025
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| author | Bisterzo, Andrea Savo, Alessandro |
| author_facet | Bisterzo, Andrea Savo, Alessandro |
| contents | Let $M$ be a Riemannian manifold and $Ω$ a smooth domain of $M$. We study the following heat diffusion problem: assume that the initial temperature is equal to $1$, uniformly on $Ω$, and is $0$ on its complement. Heat will then flow away from $Ω$ to its complement, and we are interested in the temperature on the boundary of $Ω$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_08389 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity of an overdetermined heat equation and minimal helicoids in space-forms Bisterzo, Andrea Savo, Alessandro Differential Geometry Analysis of PDEs 35K05, 35K15, 53A10, 53C21, 53C24, 53C42 Let $M$ be a Riemannian manifold and $Ω$ a smooth domain of $M$. We study the following heat diffusion problem: assume that the initial temperature is equal to $1$, uniformly on $Ω$, and is $0$ on its complement. Heat will then flow away from $Ω$ to its complement, and we are interested in the temperature on the boundary of $Ω$ 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. |
| title | Rigidity of an overdetermined heat equation and minimal helicoids in space-forms |
| topic | Differential Geometry Analysis of PDEs 35K05, 35K15, 53A10, 53C21, 53C24, 53C42 |
| url | https://arxiv.org/abs/2507.08389 |