Related papers: Minimal hypersurfaces asymptotic to Simons cones
We partially resolve a conjecture of Meeks on the asymptotic behavior of minimal surfaces in $\mathbb{R}^3$ with quadratic area growth.
We exhibit the first set of examples of non-bumpy metrics on the $(n+1)$-sphere ($2\leq n\leq 6$) in which the varifold associated with the two-parameter min-max construction must be a multiplicity-two minimal $n$-sphere. This is proved by…
We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected. Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon. We further show that each…
We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $\pi$. We…
We prove that in a closed Riemannian manifold with dimension between $3$ and $7$, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case…
We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C*-algebras with homotopy classes of asymptotic homomorphisms as morphisms. We…
We show that a closed weakly-monotone symplectic manifold of dimension $2n$ which has minimal Chern number greater than or equal to $n+1$ and admits a Hamiltonian toric pseudo-rotation is necessarily monotone and its quantum homology is…
After quick survey of some key results and open questions about the structure of singularities of minimal surfaces, we discuss recent work~\cite{Sim23} on singularities of stable minimal hypersurfaces, including some simplifications of the…
In the present paper, we discuss the singular minimal surfaces in a Euclidean 3-space R^{3} which are minimal. In fact, such a surface is nothing but a plane, a trivial outcome. However, a non-trivial outcome is obtained when we modify the…
We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that the validity of Bernstein's theorem in dimension $n+1$ is a consequence of…
We find and classify the simplest ${\cal N}=2$ SUSY multiplets on AdS$_4$ which contain partially massless fields. We do this by studying representations of the ${\cal N}=2$, $d=3$ superconformal algebra of the boundary, including new…
In this paper, we prove that a closed minimal hypersurface in $\SSS$ with $\lambda_1<n$ has Morse index at least $n+4$, providing a partial answer to a conjecture of Perdomo. As a corollary, we re-obtain a partial proof of the famous Urbano…
E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the…
The quantum nonrelativistic spin-1/2 planar systems in the presence of a perpendicular magnetic field are known to possess the N=2 supersymmetry. We consider such a system in the field of a magnetic vortex, and find that there are just two…
Motivated by problems on apparent horizons in general relativity, we prove the following theorem on minimal surfaces: Let $g$ be a metric on the three-sphere $S^3$ satisfying $Ric(g) \geq 2 g$. If the volume of $(S^3, g)$ is no less than…
We find the first examples of triply periodic minimal surfaces of which the intrinsic symmetries are all of horizontal type.
A well-known conjecture of Yau states that the area of one of Clifford minimal hypersurfaces $S^k\big{(}\sqrt{\frac{k}{n}}\, \big{)}\times S^{n-k}\big{(}\sqrt{\frac{n-k}{n}}\, \big{)}$ gives the lowest value of area among all non-totally…
We prove that there are no pseudoholomorphic theories of anything other than curves, even if one allows more general spaces than almost complex manifolds. The proof is elementary, except for theories of pseudoholomorphic hypersurfaces,…
We show generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, we prove that for any metric $g$ in a $C^{\infty}$-generic subset of the family of complete metrics which are thick…
Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of $-2\pi$.…