An official website of the United States government
Abstract We construct nonlinear entire anisotropic minimal graphs over$$\mathbb{R}^{4}$$ , completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over$$\mathbb{R}^{n},\, n \geq 8$$ .
more »
« less
This content will become publicly available on March 20, 2026
Stability of the catenoid for the hyperbolic vanishing mean curvature equation outside symmetry
Abstract We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the$$n$$ -dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for$$n \geq 5$$ . The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution.
more »
« less
- Award ID(s):
- 1945615
- PAR ID:
- 10586489
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Inventiones mathematicae
- ISSN:
- 0020-9910
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We consider the Boltzmann equation in a convex domain with a non-isothermal boundary of diffuse reflection. For both unsteady/steady problems, we construct solutions belonging to$$W^{1,p}_x$$ for any$$p<3$$ . We prove that the unsteady solution converges to the steady solution in the same Sobolev space exponentially quickly as$$t \rightarrow \infty $$ .more » « less
-
Abstract The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${{\mathbb {R}}}^{d+2}$$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\ge 4$$ .more » « less
-
Abstract We investigate the well-posedness of a class of stochastic second-order in time damped evolution equations in Hilbert spaces, subject to the constraint that the solution lies within the unitary sphere. Then, we focus on a specific example, the stochastic damped wave equation in a bounded domain of ad-dimensional Euclidean space, endowed with the Dirichlet boundary condition, with the added constraint that the$$L^2$$ -norm of the solution is equal to one. We introduce a small mass$$\mu >0$$ in front of the second-order derivative in time and examine the validity of a Smoluchowski–Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-Itô correction term.more » « less
-
Abstract The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${\mathbb {R}}^{d+2}$$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions$$d \geqq 4$$ . In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\geqq 2$$ . This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.more » « less
