arXiv subjects

Pam Sargent

Publications and source records attributed to Pam Sargent.

4 recordsLinked to original sources

arXiv2019

Existence and classification of $S^1$-invariant free boundary annuli and Möbius bands in $\mathbb{B}^n$

We explicitly classify all $S^1$-invariant free boundary minimal annuli and Möbius bands in $\mathbb{B}^n$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $S^1$-invariant metrics on the annulus and Möbius band. First, we determine the supremum of the $k$-th normalized Steklov eigenvalue among all $S^1$-invariant metrics on the Möbius band for each $k \geq 1$, and show that it is achieved by the induced metric from a free boundary minimal embedding of the Möbius band into $\mathbb{B}^4$ by $k$-th Steklov eigenfunctions. We then show that the critical metrics of the normalized Steklov eigenvalues on the space of $S^1$-invariant metrics on the annulus and Möbius band are the induced metrics on explicit free boundary minimal annuli and Möbius bands in $\mathbb{B}^3$ and $\mathbb{B}^4$, including some new families of free boundary minimal annuli and Möbius bands in $\mathbb{B}^4$. Finally, we prove that these are the only $S^1$-invariant free boundary minimal annuli and Möbius bands in $\mathbb{B}^n$.

math.DG↗

arXiv2017

Existence of harmonic maps into CAT(1) spaces

Let $φ\in C^0 \cap W^{1,2}(Σ, X)$ where $Σ$ is a compact Riemann surface, $X$ is a compact locally CAT(1) space, and $W^{1,2}(Σ,X)$ is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map $u:Σ\to X$ homotopic to $φ$ or there exists a conformal harmonic map $v:\mathbb S^2 \to X$. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.

math.DG↗

arXiv2016

Regularity of Harmonic Maps from Polyhedra to CAT(1) Spaces

We determine regularity results for energy minimizing maps from an $n$-dimensional Riemannian polyhedral complex $X$ into a CAT(1) space. Provided that the metric on $X$ is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the $(n-2)$-skeleton, we improve the regularity to locally Lipschitz. Finally, for points $x \in X^{(k)}$ with $k \leq n-2$, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of $x \in X$.

math.DG↗

arXiv2016

Index bounds for free boundary minimal surfaces of convex bodies

In this paper, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in $\mathbb{R}^3$ tends to infinity as its genus or the number of boundary components tends to infinity.

math.DG↗