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Aria Halavati

Publications and source records attributed to Aria Halavati.

5 recordsLinked to original sources
arXiv2026

Min-Max Construction of Anisotropic Minimal Surfaces with Genus Bound

We establish an anisotropic analogue of the celebrated theorem of Meeks-Simon-Yau: every minimizing sequence of surfaces within a fixed isotopy class converges to a smooth stable anisotropic minimal surface, with genus lower semicontinuity. This result also strengthens White's foundational existence theory for anisotropic minimal disks. As an application, we develop an anisotropic Simon-Smith min-max theory. In every closed $3$-manifold, we construct anisotropic min-max sequences within fixed isotopy classes whose limits are stable anisotropic minimal surfaces that are smooth except possibly at a single point. If the integrand satisfies either an ellipticity bound or a $C^3$-pinching condition, we remove the singular point by proving two independent removable singularity theorems for anisotropic minimal surfaces that are smooth and stable away from finitely many points. These removable singularity results also allow to remove the singularities arising in the anisotropic Almgren-Pitts min-max construction in $3$-manifolds of De Philippis-De Rosa and in its multiparameter variants.

math.DG↗
arXiv2024

Decay of excess for the abelian Higgs model

In this article we prove that entire critical points $(u,\nabla)$ of the self-dual $U(1)$-Yang-Mills-Higgs functional $E_1$, with energy $$E_1(u,\nabla;B_R):=\int_{B_R}\left[|\nabla u|^2+\frac{(1-|u|^2)^2}{4}+|F_\nabla|^2\right]\leq(2\pi+\tau(n)) \omega_{n-2}R^{n-2}$$ for all $R>0$, have unique blow-down. Moreover, we show that they are two-dimensional in ambient dimension $2\leq n\leq4$, or in any dimension $n\ge2$ assuming that $(u,\nabla)$ is a local minimizer, thus establishing a co-dimension-two analogue of Savin's theorem. The main ingredient is an Allard-type improvement of flatness.

math.AP↗
arXiv2023

New weighted inequalities on two-manifolds

We establish a new class of weighted $L^2$ Poincar\'e and elliptic functional inequalities on smooth two-manifolds with explicit constants, for a family of weights satisfying a differential equation. This family includes, in particular, weights comparable to products of positive powers of the geodesic distance to finitely many points. Our primary motivation is the derivation of estimates associated with a weighted Hodge decomposition for one-forms.

math.AP↗
arXiv2023

Quantitative stability of Yang-Mills-Higgs instantons in two dimensions

We prove that if an N-vortex pair nearly minimizes the Yang-Mills-Higgs energy, then it is second order close to a minimizer. First we use new weighted inequalities in two dimensions and compactness arguments to show stability for sections with some regularity. Second we define a selection principle using a penalized functional and by elliptic regularity and smooth perturbation of complex polynomials, we generalize the stability to all nearly minimizing pairs. With the same method, we also prove the analogous second order stability for nearly minimizing pairs on nontrivial line bundles over arbitrary compact smooth surfaces.

math.AP↗