We provide a proof of effective uniformization for nearly round 2-spheres, utilizing an identity related to the third-order differential of the conformal factor. This identity is connected to the geometry of the embedded spacelike surface within the Minkowski lightcone. Additionally, we investigate the stability of the effective uniformization introduced by Klainerman and Szeftel. Our proof is based on a geometric insight: an isometric embedding of a round sphere into Euclidean space can be constructed using an orthogonal basis of the first eigenspace of the Laplacian operator, with the rectangular coordinates corresponding to the basis functions. By adopting these approaches, we simplify both the proofs of effective uniformization and its stability, while also refining the assumptions underlying both results.
Let $S$ be a surface with a metric $d$ satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into $(S,d)$ is a branched covering. As a consequence, if $(S,d)$ is homeomorphically equivalent to the 2-sphere $\mathbb S^2$, then it is conformally equivalent to $\mathbb S^2$.
With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discrete uniformization via hyper-ideal circle patterns always exists and is unique. We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.
This is a follow-up of our paper \cite{KS-Kerr1} on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of $\ell=1$ modes on a $2$-sphere embedded in a $1+3$ vacuum manifold. This is based on a new, effective, version of the classical uniformization theorem which allows us to define such modes and prove their stability for spheres with comparable metrics. The reformulation allows us to prove a second, in…
In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow. Such an analytic approach also sheds light on how to obtain the boundedness for E_1 energy in the study of general Kähler manifolds.
It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface M\subset S^{n+1} with cmc cannot take the values 1,2,3... n.
We prove that for every $\Q$-homological Finsler 3-sphere $(M,F)$ with a bumpy and irreversible metric $F$, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics.
In this short note we extend an estimate due to J. Simons on the first stability eigenvalue of minimal hypersurfaces in spheres to the singular setting. Specifically, we show that any singular minimal hypersurface in $S^{n+1}$, which is not totally geodesic and satisfies the α-structural hypothesis, has first stability eigenvalue at most -2n, with equality if and only if it is a product of two round spheres. The equality case was settled independently in the classical setting by Wu and Perdomo.
Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,α}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.
It is a well known fact that, if $Σ$ is an Einstein hypersurface with positive scalar curvature, then it is a round sphere. We give a stable version of this result showing that if a hypersurface is almost Einstein in a $L^p$-sense, then it is $W^{2, \, p}$ - close to the round sphere. The result is given in a quantitative way.
We show that if a closed surface in $\mathbb{R}^3$ has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.
In this paper we discuss the stability of geodesic spheres in $\mathbb{S}^{n+1}$ under constrained curvature flows. We prove that under some standard assumptions on the speed and weight functions, the spheres are stable under perturbations that preserve a volume type quantity. This extends results by Escher and Simonett, 1998, and the author, 2015, to a Riemannian manifold setting.
The rigidity theorems of Llarull and Marques-Neves, which show two different ways scalar curvature can characterize the sphere, have associated stability conjectures. Here we produce the first examples related to these stability conjectures. The first set of examples demonstrates the necessity of including a condition on the minimum area of all minimal surfaces to prevent bubbling along the sequence. The second set of examples constructs sequences that do not converge in the Gromov-Hausdorff sense but do converge in the volume preserving intrinsic flat sense. In order to construct such sequenc…
We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the $L^2$-sense, then the flow exists for all time and converges exponentially to an umbilical sphere.
In this paper, we prove that for every Finsler $4$-dimensional sphere $(S^4,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\frac{25}{9}\left(\fracλ{1+λ}\right)^2<K\le 1$ with $λ<\frac{3}{2}$, either there exist at least four prime closed geodesics, or there exist exactly three prime non-hyperbolic closed geodesics and at least two of them are irrationally elliptic.
Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $κ$. For any point $p\in M$ and any number $λ\in\mathbb{R}$ with $λ^2+κ>0$, we show existence of a $C^2$ spherical surface $\mathcal{S}_λ(p)$ immersed in $M$ with constant mean curvature $λ$. Our construction recovers in particular the description of Pansu spheres in the first Heisenberg group and the sub-Riemannian $3$-sphere. Then, we study variational properties of $\mathcal{S}_λ(p)$ related to the area functional. First, we obtain uniqueness results for the spheres $\mathcal{S}_λ(p)$ as critical…
We establish various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres in symplectic $4-$manifolds with $κ=-\infty$. This involves the explicit construction of spheres in rational manifolds via a new construction technique called the tilted transport.
We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its $C^2$-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.
The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,\ldots,k_{n-1})$ satisfying suitable conditions. In this paper we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator $f$ is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is quantified in terms of t…
This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for $|a|\ll m$. The paper builds on the strategy laid out in \cite{KS} in the context of the nonlinear stability of Schwarzschild for axially symmetric polarized perturbations. In fact the central idea of \cite{KS} was the introduction and construction of generally covariant modulated (GCM) hypersurfaces on which specific geometric quantities take Schwarzschildian values. This was made possible by taking into account the full general covariance of the Einstein vacuum equatio…
The Helfrich functional, denoted by H^{c_0}, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H^{c_0} among all possible configurations. The functional integrates a spontaneous curvature parameter c_0 together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to repre…
