We establish a sharp bound on the Hausdorff dimension of the non-branch singular set of branched stable minimal immersed hypersurfaces whose singular sets have locally finite $\mathcal H^{n-2}$-measure: the non-branch singular set is empty when $n=2$, discrete when $n=3$, and has Hausdorff dimension at most $n-3$ when $n\geq4$. We also construct a non-flat stable minimal cone in $\mathbb R^4$ arising from a branched minimal immersion whose vertex is a non-branch singularity. Taking products with Euclidean factors yields examples whose non-branch singular sets have Hausdorff dimension exactly $n-3$, showing that our regularity bound is sharp in every dimension $n\geq3$. The main ingredients in our proof are a generalized Schoen inequality and a corresponding branched sheeting theorem near stationary classical cones and unions of hyperplanes.
Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption an...
Let $(X,d,p)$ be a noncollapsed pointed Gromov--Hausdorff limit of complete $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, where $n\ge4$. We assume that, on each bounded ball, the integral of the $3/2$ power of the small-ball volume deficit relative to the hyperbolic comparison volume...
Let $(X,d,p)$ be an $n$-dimensional noncollapsed Ricci limit space, where $n\ge4$. Under an integral volume-deficit bound, we prove that the metric singular set has Hausdorff codimension at least four and sigma-finite $(n-4)$-dimensional Hausdorff measure. This establishes a special case of the codimension-four regular...
We prove a sharp partial regularity result for Hamiltonian stationary Lagrangian Lipschitz submanifolds in arbitrary smooth almost K\"ahler manifolds: every weak solution of the corresponding equation is smooth away from a relatively closed singular set of Hausdorff dimension at most $n-5$. We show that the estimate is...
Arunima Bhattacharya, Gerard Orriols, A. Skorobogatova· 2 citations· ⚡1
We show that every closed, nonorientable surface can be minimally embedded in $\mathbb{S}^4$, providing in particular the first known examples of embedded, nonorientable minimal surfaces in $\mathbb{S}^4$ with negative Euler characteristic. All of the surfaces we obtain have area below $8\pi$, which has applications to...
Mikhail Karpukhin, R. Kusner, Peter McGrath et al.· 0 citations
Let $V$ be a complex symplectic vector space and let $\mathbb{P}(V)$ carry the induced contact structure and symplectic polarity. We study the loci of hypersurfaces containing codimension-two cones supported on hyperplanes. The starting point is the following characterization: a projective subvariety $X\subset\mathbb{P...
Gabriel Guedes· 0 citations
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