arXiv subjects

Bryan Dimler

Publications and source records attributed to Bryan Dimler.

5 recordsLinked to original sources

arXiv2026

Special Lagrangians with multiple isolated singularities

We extend the Caffarelli-Hardt-Simon perturbation argument for truncated regular minimal cones to the special Lagrangian setting and prove a bridge principle for regular special Lagrangian cones in the spirit of Nathan Smale. Our bridge principle yields a general existence theorem for conically singular special Lagrangian submanifolds with prescribed regular tangent cones: for any finite list of such cones in $\mathbb{C}^m$ having the same Lagrangian angle and suitably arranged, there exists a connected special Lagrangian submanifold with boundary and isolated conical singularities whose tangent cones at its singularities are precisely the prescribed cones. In particular, we obtain new special Lagrangian submanifolds in $\mathbb{C}^m$ with multiple prescribed isolated conical singularities.

math.DG↗

arXiv2025

Uniqueness in the Plateau problem for calibrated currents

We show that every compactly supported smoothly calibrated integral current with connected $C^{3,\alpha}$ boundary is the unique solution to the oriented Plateau problem for its boundary data. The same holds true for compactly supported $``$continuously calibrated$"$ integral flat chains. This is proved as a consequence of the boundary regularity theory for area-minimizing currents and a unique continuation argument in the spirit of Frank Morgan. In codimension one, the argument yields a sufficient condition for uniqueness in the oriented Plateau problem expressed in terms of the regularity of the calibrating form.

math.DG↗

arXiv2024

Minimal submanifolds with multiple isolated singularities

We extend Smale's singular bridge principle [Ann. of Math. 130 (1989), 603-642] for $n$-dimensional strictly stable minimal cones in $\mathbb{R}^{n+1}$ $(n \geq 7$) to arbitrary codimension and each $n \geq 3$. We then apply the procedure to copies of the Lawson-Osserman cone to produce a four dimensional minimal graph in $\mathbb{R}^7$ with any finite number of isolated singularities.

math.DG↗

arXiv2024

Strict stability of calibrated cones

We study the strict stability of calibrated cones with an isolated singularity. For special Lagrangian cones and coassociative cones, we prove the strict stability. In the complex case, we give non-strictly stable examples.

math.DG↗

arXiv2023

Partial regularity for Lipschitz solutions to the minimal surface system

In this paper, we study the regularity of several notions of Lipschitz solutions to the minimal surface system with an emphasis on partial regularity results. These include stationary solutions, integral weak solutions, and viscosity solutions. We also prove an interior gradient estimate for classical solutions to the system using the maximum principle, assuming the area-decreasing condition and that all but one component have small infinity norm.

math.AP↗