Minimal & CMC surfaces

We describe the self-expanders of the mean curvature flow in ${\mathbb{R}}^{p+q+2}$ invariant under $O(p+1)\times O(q+1)$. Apart from the Clifford minimal cone, every such hypersurface is complete, proper and noncompact, and belongs to one of three families: two one-parameter families of embedded hypersurfaces diffeomorphic to ${\mathbb{R}}^{p+1}\times {\mathbb{S}}^q$ or to ${\mathbb{R}}^{q+1}\times {\mathbb{S}}^p$, whose profile curves leave an axis, and a two-parameter family diffeomorphic to $\mathbb{R}\times {\mathbb{S}}^p\times {\mathbb{S}}^q$, parametrized by the point of the profile curve closest to the origin, which is the generic family: its set of initial conditions is open and of full measure in the phase space. Every end is asymptotically conical, with explicit decay rates, and every such hypersurface has Euclidean volume growth. A profile curve of the third family has at most one loop, and the point of the curve closest to the origin lies on that loop.

We describe the global structure of the self-shrinkers of the mean curvature flow in ${\mathbb{R}}^{p+q+2}$ invariant under $O(p+1)\times O(q+1)$, without assuming properness or embeddedness. The main tool is a pair of piecewise monotone quantities along the profile curves, in the spirit of Hsiang's analysis of constant mean curvature hypersurfaces. We prove that every such self-shrinker other than the two invariant cylinders intersects both of them transversely, and all of them except the Clifford minimal cone are complete. Apart from those of the cylinders, each end is either asymptotic to a cone or meets both cylinders infinitely many times. Consequently, a complete noncompact example is proper if and only if all its ends are asymptotically conical, and this is equivalent to polynomial volume growth, to finite entropy, and to finitely many intersections with the cylinders. The reduced flow preserves a finite measure, and for almost every initial condition the profile curve is either closed or generates a complete nonproper self-shrinker of infinite entropy. We also bound the winding of the profile curve by its number of cylinder crossings and localize its self-intersections.

We study complete translating solitons for the mean curvature flow in arbitrary codimension under quadratic pinching of the second fundamental form. We prove a codimension reduction theorem at the Andrews-Baker threshold: if a complete $n$-dimensional connected translator isometrically immersed in $({\mathbb{R}}^{n+p},⟨\cdot ,\cdot ⟩)$ satisfies $$|B{|}^2\le \left(\frac{4}{3n}-{\epsilon }_0\right)|H{|}^2$$ for some ${\epsilon }_0>0$, then either it is an affine $n$-plane or $|H|>0$ everywhere and the translator is contained in an $(n+1)$-dimensional affine subspace. In particular, this improves in low dimensions the pinching range previously obtained from codimension reduction results for ancient mean curvature flows. Our proof is purely elliptic: viewing translators as weighted minimal submanifolds, we combine Simons-type identities for the drift Laplacian with refined gradient estimates and an Omori-Yau maximum principle. As a consequence, under the stronger pinching condition $$|B{|}^2\le (c_n-{\epsilon }_0)|H{|}^2,c_n=\min \left\{\frac{4}{3n},\frac{1}{n-2}\right\}$$ for $n\ge 3$, with $c_n=2/3$ when $n=2$, we obtain a rigidity theorem: the translator is either an affine plane or a bowl soliton contained in an $(n+1)$-dimensional affine subspace. No entropy assumption is required.

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n+m\ge 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\text{singular minimal cone as the asymptotic model}$.

We study oriented surfaces in the Heisenberg space ${\mathrm{Nil}}_3$ whose mean curvature $H$ at each point is $H=⟨N,{\partial }_z⟩+\lambda$, where $N$ is the unit normal, ${\partial }_z$ is the vertical Killing vector field and $\lambda \in \mathbb{R}$. These surfaces are known as $\lambda$-translators and generalize, among others, minimal and positive constant mean curvature surfaces, and also translating solitons of the mean curvature flow. The objective in this paper is to classify $\lambda$-translators invariant by the following one-parameter groups of isometries of ${\mathrm{Nil}}_3$: left-translations, rotations and helicoidal motions.

We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.

We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading in particular to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.

In this work, we discuss the stability of Donaldson's flow of surfaces in a hyperkähler 4-manifold. In, Wang and Tsai proved a uniqueness theorem and $C^1$ dynamic stability theorem of the mean curvature flow for minimal surface. We extend their results and obtain a similar dynamic stability theorem of the hyperkähler flow.

We consider the hyperbolic three-space in the half-space model endowed with a metric Lie group structure. In this setting, translation surfaces are defined as products of two curves $\alpha$ and $\beta$ with respect to the Lie group operation. We investigate minimal surfaces and solitons to the mean curvature flow arising from specific types of products of these curves. In particular, we provide classification results for minimal surfaces, hyperbolic translators, and conformal solitons to the mean curvature flow.

In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.

This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic $3$-manifolds of finite volume, particularly its comparison to the hyperbolic minimal surface entropy in terms of sectional and scalar curvature. On one hand, for metrics that are bilipschitz equivalent to the hyperbolic metric and have sectional curvature bounded above by $-1$ and uniformly bounded below, we show that the entropy achieves its minimum if and only if the metric is hyperbolic. On the other hand, by analyzing the convergence rate of the Ricci flow toward the hyperbolic metric, we prove that among all metrics with scalar curvature bounded below by $-6$ and with non-positive sectional curvature on the cusps, the entropy is maximized at the hyperbolic metric, provided that it is infinitesimally rigid. Furthermore, if the metrics are uniformly $C^0$-close to the hyperbolic metric and asymptotically cusped, then the entropy associated with the Lebesgue measure is uniquely maximized at the hyperbolic metric.

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric $h_0$, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to $h_0$ in a weighted Hölder norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].

We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. We introduce a structural condition, obtained from a second-order ordinary differential equation, which allows the construction of explicit boundary barriers and it can also be applied to unbounded domains. In the setting of Hadamard manifolds, this condition relates the geometry of the domain to the admissible boundary data in a direct way. In Euclidean space, the condition leads to solvability under geometric hypotheses of a different nature from those in the classical Jenkins-Serrin theory, and in some configurations it applies where the Jenkins-Serrin method does not. A central point of the present approach is that the geometric restrictions and the boundary data enter independently. The same barrier construction can be used for graphical mean curvature flow. This yields short-time existence with prescribed boundary values even when the boundary of the domain is not mean convex. When mean convexity is present, one recovers the classical graphical setting.

From the perspective of Morse theory, it is natural to investigate gradient flow trajectories between critical points. In this short note, we explore the minimal hypersurface analogue of this phenomenon and present examples that suggest additional topological and variational obstructions to the existence of connecting mean curvature flows.

In this paper, we propose a new assumption (1.2) that involves a small oscillation and $C^2$ norms for maps from smooth bounded domains into Euclidean spaces. Furthermore, by assuming that the domain has non-negative Ricci curvature, we establish the Dirichlet problem for the minimal surface system via the mean curvature flow (MCF) with boundary. The long-time existence of such flow is derived using Bernstein-type theorems of higher codimensional self-shrinkers in the whole space and the half-space. Another novel aspect is that our hypothesis imposes no restriction on the diameter of the domains, which implies an existence result for an exterior Dirichlet problem of the minimal surface system.

In this work we study the existence of solutions to the Mean Curvature Flow for which the initial condition has the structure of a two-dimensional Lie subgroup within a Lie group of dimension three. We consider Lie groups with a fixed left-invariant metric and first observe that if the Lie group is unimodular, then every Lie subgroup is a minimal surface (hence a trivial solution). For this reason we focus on non-unimodular Lie groups, finding the evolution of every Lie subgroup of dimension 2 (within a 3 dimensional Lie group). These evolutions are self-similar for abelian subgroups (i.e. evolve by isometries), but not self-similar in the other cases.

In this paper, we study the drifted Laplacian ${\Delta }_f$ on a hypersurface $M$ in a Ricci shrinker $(\overline{M},g,f)$. We prove that the spectrum of ${\Delta }_f$ is discrete for immersed hypersurfaces with bounded weighted mean curvature in a Ricci shrinker with a mild condition on the potential function. Next, we give a lower bound for the first nonzero eigenvalue of ${\Delta }_f$ when the hypersurface is an embedded $f$-minimal one. This estimate contains the case of compact minimal hypersurfaces in a positive Einstein manifold, in particular Choi and Wang's estimate for minimal hypersurfaces in a round sphere. The estimate also recovers the ones of Ding-Xin and Brendle-Tsiamis on self-shrinkers.

In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in a shrinking gradient Ricci soliton $(M^n,g,f)$ with scalar curvature $R\ge (n-1)\lambda$ must have vanished second fundamental form and vanished normal Ricci curvature. For shrinking gradient Ricci solitons with scalar curvature $R\ge (n-1)\lambda$, the existence of an area-minimizing hypersurface would imply that $M$ is splitting.

We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let $(M^3,g)$ be an asymptotically flat manifold with scalar curvature $R\ge 0$. Then, for each constant $c>0$, there exists a compact, almost-embedded, free boundary constant mean curvature surface $\Sigma \subset M$ with mean curvature $c$. Likewise, let $(M^3,g)$ be an asymptotically hyperbolic manifold with scalar curvature $R\ge -6$. Then, for each constant $c>2$, there exists a compact, almost-embedded, free boundary constant mean curvature surface $\Sigma \subset M$ with mean curvature $c$. The proof combines min-max theory with the following fact about inverse mean curvature flow which is of independent interest: for any $T$ the inverse mean curvature flow emerging out of a point $p$ far enough out in an asymptotically flat (or asymptotically hyperbolic) end will remain smooth for all times $t\in (-\infty ,T]$.