arXiv subjects

Elham Matinpour

Publications and source records attributed to Elham Matinpour.

6 recordsLinked to original sources

arXiv2026

Free Boundary Plateau Model Cones in $\mathbb{B}^n$ are Rigid under Conformal Minimal Immersions

The classical theorem of Nitsche asserts that every free-boundary minimal disk in the unit ball $\mathbb{B}^3$ is an equatorial flat disk. Fraser and Schoen later generalized this rigidity theorem to arbitrary dimensions and ambient spaces of constant sectional curvature. In previous work, the author established an analogous rigidity result for the singular $Y$-cone: any conformal free-boundary minimal immersion of the flat $Y$-cone into $\mathbb{B}^n$ is congruent to the flat $Y$-cone. In this paper we treat the remaining classical two-dimensional Plateau singularity model, namely the tetrahedral $T$-cone. We prove that every conformal free-boundary minimal immersion of the flat $T$-cone into $\mathbb{B}^n$ is congruent, up to an orthogonal transformation, to the flat $T$-cone itself. As a consequence, combining this result with the Nitsche--Fraser--Schoen theorem and the previously established $Y$-cone rigidity theorem, we obtain a unified rigidity theorem for the classical Plateau model domains: any free-boundary minimal Plateau surface in $\mathbb{B}^n$ conformal to a plane disk, a $Y$-cone, or a $T$-cone must be congruent to the corresponding model.

math.DG↗

arXiv2025

First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in $\mathbb{R}^3$

Consider $\mathbb{R}^3$ equipped with the Euclidean metric and the Gaussian measure. Let $\Sigma$ be a complete embedded self-shrinker in $\mathbb{R}^3$ with the induced metric and weighted measure, and let $\lambda_1$ denote the first eigenvalue of the drift Laplacian in the weighted $L^2$ space. Inspired by Choe and Soret's estimate of the first eigenvalue of the Laplacian on symmetric minimal surfaces in $\mathbb{S}^3$, we prove that $\lambda_1$= 1/2 for self-shrinkers invariant under the dihedral group $\mathbb{D}_{g+1}$ or the prismatic group $\mathbb{D}_{g+1}\times \mathbb{Z}_2$. In particular, this holds for known self-shrinkers confirming a universal spectral property tied to their symmetry.

math.DG↗

arXiv2025

Rigidity and index of free boundary minimal Y-cones in the unit ball

J. C. C. Nitsche proved that any minimal disk satisfying the free boundary condition in the unit ball $B^3$ must be an equatorial flat disk. Later, Fraser and Schoen extended this rigidity theorem to higher dimensions and to ambient spaces of constant curvature. In this paper, we establish an analogue of the Nitsche-Fraser-Schoen theorem for singular free boundary minimal surfaces of Y-type in $B^n$. Specifically, we prove that any conformal and minimal immersion of the standard compact flat Y-cone meeting $\partial B^n$ orthogonally must itself be a flat Y-cone. In addition, we compute the Morse index of the free boundary flat Y-cone and show that it equals $2(n-2)$. Furthermore, we prove that Y-cones are the only free boundary minimal surfaces in $B^n$ with Morse index $2(n-2)$.

math.DG↗

arXiv2025

Rigidity of Minimal Surfaces with Y-singularities and low Morse Index in $\mathbb{R}^3$

We investigate the geometric constraints imposed by low Morse index on minimal surfaces with Y-singularities, focusing on the classification of those with Morse index one. Our rigidity result establishes a partial uniqueness theorem, highlighting the Y-catenoid as a distinguished example among complete, two-sided minimal surfaces in $\mathbb{R}^3$ with Y-singularities and Morse index one.

math.DG↗

arXiv2024

Morse index of y-singular minimal surfaces

In this paper, we compute the Morse index of rotationally symmetric minimal Y-singular surfaces under assumption that the Y-singularities form a single circle. This computation is carried out by utilizing information from two simpler problems: the first deals with the fixed boundary problem on singularities, and the second focuses on the Dirichlet-to-Neumann map associated with the stability operator. Notably, our findings reveal that the index of the Y -catenoid is one.

math.DG↗

arXiv2024

On the Area of Immersed Minimal Annuli in a Slab

We organize minimal annuli in a slab based on the winding number of the circles that foliate them and study the area of minimal annuli with given winding number. Specifically, we deduce some results regarding the convexity of the length function of corresponding level curves, and apply them to estimate the area of the annuli by comparing to the area of waist of covers of catenoids.

math.DG↗