Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFree boundary minimal surfaces with connected boundary and arbitrary genus
81
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We employ min-max techniques to show that the unit ball in $mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
rate research
Read More
In this paper we establish a connection between free boundary minimal surfaces in a ball in $mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.
We adapt the viscosity method introduced by Rivi`ere to the free boundary case. Namely, given a compact oriented surface $Sigma$, possibly with boundary, a closed ambient Riemannian manifold $(mathcal{M}^m,g)$ and a closed embedded submanifold $mathcal{N}^nsubsetmathcal{M}$, we study the asymptotic behavior of (almost) critical maps $Phi$ for the functional begin{align*} &E_sigma(Phi):=operatorname{area}(Phi)+sigmaoperatorname{length}(Phi|_{partialSigma})+sigma^4int_Sigma|{mathrm {I!I}}^Phi|^4,operatorname{vol}_Phi end{align*} on immersions $Phi:Sigmatomathcal{M}$ with the constraint $Phi(partialSigma)subseteqmathcal{N}$, as $sigmato 0$, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection $mathcal{F}$ of compact subsets of the space of smooth immersions $(Sigma,partialSigma)to(mathcal{M},mathcal{N})$, assuming $mathcal{F}$ to be stable under isotopies of this space we show that the min-max value begin{align*} &beta:=inf_{Ainmathcal{F}}max_{Phiin A}operatorname{area}(Phi) end{align*} is the sum of the areas of finitely many branched minimal immersions $Phi_{(i)}:Sigma_{(i)}tomathcal{M}$ with $partial_ uPhi_{(i)}perp Tmathcal{N}$ along $partialSigma_{(i)}$, whose (connected) domains $Sigma_{(i)}$ can be different from $Sigma$ but cannot have a more complicated topology. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_sigma$ and, along the way, we simplify several arguments from the original work. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.
In this note, we study minimal Lagrangian surfaces in $mathbb{B}^4$ with Legendrian capillary boundary on $mathbb{S}^3$. On the one hand, we prove that any minimal Lagrangian surface in $mathbb{B}^4$ with Legendrian free boundary on $mathbb{S}^3$ must be an equatorial plane disk. One the other hand, we show that any annulus type minimal Lagrangian surface in $mathbb{B}^4$ with Legendrian capillary boundary on $mathbb{S}^3$ must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng (Sci. China Math., 2020).
Given a $C^k$-smooth closed embedded manifold $mathcal Nsubset{mathbb R}^m$, with $kge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $partial S eqemptyset$, we consider $frac 12$-harmonic maps $uin H^{1/2}(partial S,mathcal N)$. These maps are critical points of the nonlocal energy begin{equation}E(f;g):=int_Sbig| ablawidetilde ubig|^2,dtext{vol}_g,end{equation} where $widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $frac 12$-energies at each boundary component of $partial S$ (suitably identified with the circle $mathcal S^1$), plus a quadratic term which is continuous in the $H^s(mathcal S^1)$ topology, for any $sinmathbb R$. We show the $C^{k-1,delta}$ regularity of $frac 12$-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichmuller space of $S$.
In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the genus and number of boundary components.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
