A mean curvature type flow with capillary boundary in a unit ball
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Calculus of Variations
A mean curvature type flow with capillary boundary in a unit ball Guofang Wang1 · Liangjun Weng1,2 Received: 10 January 2020 / Accepted: 30 June 2020 © The Author(s) 2020
Abstract In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional E. We show that it has the longtime existence and subconverges to spherical caps. As an application, we solve an isoperimetric problem for hypersurfaces with capillary boundary. Mathematics Subject Classification Primary: 53C44; Secondary: 35K93
1 Introduction In this paper, we are interested in a mean curvature type flow in the unit ball Bn+1 ⊂ Rn+1 with capillary boundary. Roughly speaking, given a Riemannian manifold N n+1 with a smooth boundary ∂ N , a hypersurface with capillary boundary in N is an immersed hypersurface which intersects ∂ N at a constant contact angle θ ∈ (0, π). For closed hypersurfaces, the mean curvature flow plays an important role in geometric analysis and has been extensively studied. One of classical results proved by Huisken [19] states that it contracts a closed convex hypersurface into a round point. Mean curvature type flows with a constraint play an important role in the study of isoperimetric problems. The following curve-shortening (and area-preserving) flow was studied by Gage [10]. Let
Communicated by J. Jost.
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Guofang Wang [email protected] Liangjun Weng [email protected]; [email protected]
1
Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg im Breisgau 79104, Germany
2
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, People’s Republic of China 0123456789().: V,-vol
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G. Wang, L. Weng
γ : S1 × [0, T ) → R2 satisfy
2π ∂t γ = κ − ν, L
where κ is the geodesic curvature of γ , L is the length of the curve at scale t, and ν is the outward unit normal vector of curve γ (·, t). In a higher dimensional Euclidean space, Huisken introduced a non-local type mean curvature flow in [21]: Given a closed, connected hypersurface M, consider a family of embeddings x : M × [0, T ) → Rn+1 satisfies
∂t x = (c(t) − H )ν, H dμ
t is the average of the mean curvature H of Mt := x(M, t) and ν is the where c(t) := M|M t| unit outward normal vector field of Mt . Huisken proved that such a volume preserving flow converges to a round sphere if the initial hypersurface is uniformly convex. There has been a lot of work on such geometric flows. Here we just mention further [2] for studying such kind of flow in the case where the ambient space is Riemannian manifold and [30] for the extension to a general mixed volume preserving mean curvature flow. As one of applications, such a volume (or area)-preserving flow could be used to prove optimal geometric inequalities. In order to establish optimal geometric inequalities, there is another type
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