Sphere Theorem Research Articles

A New Differentiable Sphere Theorem and Its Applications

In this paper, we use the Lichnerowicz Laplacian to prove new results: the sphere theorem and the integral inequality for Einstein's infinitesimal deformations, which allow us to characterize spherical space forms. Our version of the sphere theorem states that a closed connected Riemannian manifold $(M, g)$ of even dimension $n>3$ is diffeomorphic to a Euclidean sphere or a real projective space if the inequality $Ric_{\rm max}(x) < n K_{\rm min}(x) g$ is true at each point $x\in M$, where $Ric_{\rm max}(x)$ is the maximum of the Ricci curvature, and $K_{\rm min}(x)$ is the minimum of the sectional curvature of $(M, g)$ at $x$. Since this inequality implies positive sectional curvature; therefore, our result partially answers Hopf's old open question.

The geometry of [formula omitted]-harmonic maps

In this paper, we motivate and extend the study of harmonic maps or Φ(1)-harmonic maps (cf. Eells and Sampson, 1964, Remark 1.3 (iii)), Φ-harmonic maps or Φ(2)-harmonic maps (cf. Han and Wei, 2022, Remark 1.3 (v)), and explore geometric properties of Φ(3)-harmonic maps by unified geometric analytic methods. We define the notion of Φ(3)-harmonic maps and obtain the first variation formula and the second variation formula of the Φ(3)-energy functional EΦ(3). By using a stress–energy tensor and the asymptotic assumption of maps at infinity, we prove Liouville type results for Φ(3)-harmonic maps. We introduce the notion of Φ(3)-Superstrongly Unstable (Φ(3)-SSU) manifold and provide many interesting examples. By using an extrinsic average variational method in the calculus of variations (cf. Wei, 1998; Wei, 1984), we find Φ(3)-SSU manifold and prove that any stable Φ(3)-harmonic maps from a compact Φ(3)-SSU manifold (into any compact Riemannian manifold) or (from any compact Riemannian manifold) into a compact Φ(3)-SSU manifold must be constant. We also prove that the homotopy class of any map from a compact Φ(3)-SSU manifold (into any compact Riemannian manifold) or (from any compact Riemannian manifold) into a compact Φ(3)-SSU manifold contains elements of arbitrarily small Φ(3)-energy. We call a compact Riemannian manifold M to be Φ(3)-strongly unstable (Φ(3)-SU) if it is not the target or domain of a nonconstant stable Φ(3)-harmonic map (from or into any compact Riemannian manifold) and also the homotopy class of any map to or from M (from or into any compact Riemannian manifold) contains elements of arbitrarily small Φ(3)-energy. We prove that every compact Φ(3)-SSU manifold is Φ(3)-SU. As consequences, we obtain topological vanishing theorems and sphere theorems by employing a Φ(3)-harmonic map as a catalyst. This is in contrast to the approaches of utilizing a geodesic (Synge, 1936), minimal surface, stable rectifiable current (Lawson Jr. and Simons, 1973; Howard and Wei, 2015; Wei, 1984), p-harmonic map (cf. Wei, 1998), etc., as catalysts. These mysterious phenomena are analogs of harmonic maps or Φ(1)-harmonic maps, p-harmonic maps, ΦS-harmonic maps, ΦS,p-harmonic maps, Φ(2)-harmonic maps, etc., (cf. Howard and Wei, 1986; Wei, 1992; Wei and Yau, 1994; Wei, 1998; Feng-Han-Li-Wei, 2021; Feng-Han-Wei, 2021).

Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres

We establish an asymptotic formula for the number of lattice points in the sets $$\begin{aligned} {\textbf{S}}_{h_1, h_2, h_3}(\lambda ): =\left\{ x\in {\mathbb {Z}}_+^3:\lfloor h_1(x_1)\rfloor +\lfloor h_2(x_2)\rfloor +\lfloor h_3(x_3)\rfloor =\lambda \right\} \quad \text {with}\quad \lambda \in {\mathbb {Z}}_+; \end{aligned}$$ where functions $$h_1, h_2, h_3$$ are constant multiples of regularly varying functions of the form $$h(x):=x^c\ell _h(x)$$ , where the exponent $$c>1$$ (but close to 1) and a function $$\ell _h(x)$$ is taken from a certain wide class of slowly varying functions. Taking $$h_1(x)=h_2(x)=h_3(x)=x^c$$ we will also derive an asymptotic formula for the number of lattice points in the sets $$\begin{aligned} {\textbf{S}}_{c}^3(\lambda ) := \{x \in {\mathbb {Z}}^3 : \lfloor |x_1|^c \rfloor + \lfloor |x_2|^c \rfloor + \lfloor |x_3|^c \rfloor = \lambda \} \quad \text {with}\quad \lambda \in {\mathbb {Z}}_+; \end{aligned}$$ which can be thought of as a perturbation of the classical Waring problem in three variables. We will use the latter asymptotic formula to study, the main results of this paper, norm and pointwise convergence of the ergodic averages $$\begin{aligned} \frac{1}{\#{\textbf{S}}_{c}^3(\lambda )}\sum _{n\in {\textbf{S}}_{c}^3(\lambda )}f(T_1^{n_1}T_2^{n_2}T_3^{n_3}x) \quad \text {as}\quad \lambda \rightarrow \infty ; \end{aligned}$$ where $$T_1, T_2, T_3:X\rightarrow X$$ are commuting invertible and measure-preserving transformations of a $$\sigma $$ -finite measure space $$(X, \nu )$$ for any function $$f\in L^p(X)$$ with $$p>\frac{11-4c}{11-7c}$$ . Finally, we will study the equidistribution problem corresponding to the spheres $${\textbf{S}}_{c}^3(\lambda )$$ .

Viability of circle and sphere theorems in potential theory and hydrodynamics via Maxwell's conjecture

AbstractIn A Treatise on Electricity and Magnetism, Maxwell determines the angles of intersection for which one may use Kelvin's inversion method to obtain the perturbed electric potential upon placing intersecting spherical conductors into a region with a known potential. There are numerous modern applications utilizing this geometric construction in potential theory and hydrodynamics, and generalized circle and sphere theorems play a foundational role in this area of mathematical physics. In his work, Maxwell gives an intuitive argument for obtaining the perturbed potential based on intersecting planar conductors and a spherical inversion, and in this paper we extend his ideas to a full proof using rotational transformations and reflections. In the process, we disprove results in [Proc Lond Math Soc., 1966:3(16)] and [Stud Appl Math., 2001:106(4); Z. Angew. Math. Mech., 2001:81(8)] on boundary value problems in hydrodynamics involving intersecting circles and spheres, and we detail the angles of intersection for which these theorems are viable. Moreover, our proof recovers a special case overlooked by Maxwell for which Kelvin's inversion method may be utilized to obtain full solutions.

Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold Mn of a complex space form M˜(4ϵ) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of Mn are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold Mn, in which Mn is homeomorphic to a sphere Sn if n≥4 and if n=3, then M3 is homotopic to a sphere S3 under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

The triangle comparison theorem for spheres

The triangle comparison theorem for spheres

Necessary and sufficient conditions for a triangle comparison theorem

We prove a version of Topogonov's triangle comparison theorem with surfaces of revolution as model spaces. Given a model surface and a Riemannian manifold with a fixed base point, we give necessary and sufficient conditions under which every geodesic triangle in the manifold with a vertex at the base point has a corresponding Alexandrov triangle in the model. Under these conditions we also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type Sphere Theorem.

Effective Results on Uniformization and Intrinsic GCM Spheres in Perturbations of Kerr

This is a follow-up of our paper (Klainerman and Szeftel in Construction of GCM spheres in perturbations of Kerr, Accepted for publication in Annals of PDE) 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, intrinsic, existence theorem for GCM spheres, expressed purely in terms of geometric quantities defined on it. A natural definition of angular momentum for such GCM spheres is also introduced, which we expect to play a key role in determining the final angular momentum for general perturbations of Kerr.

Noncritical maps on geodesically complete spaces with curvature bounded above

We define and study the regularity of distance maps on geodesically complete spaces with curvature bounded above. We prove that such a regular map is locally a Hurewicz fibration. This regularity can be regarded as a dual concept of Perelman’s regularity in the geometry of Alexandrov spaces with curvature bounded below. As a corollary, we obtain a sphere theorem for geodesically complete CAT(1) spaces.

A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere

AbstractIn this paper, we investigate topological sphere theorems for compact minimal contact CR-submanifolds of odd dimensional unit sphere. We show that if an inequality involving the warping function and the scalar curvature of the fibers is satisfied, a compact minimal contact CR-warped product submanifold of the odd dimensional unit sphere is homeomorphic to the sphere. In particular case, for 5-dimensional unit sphere, we show that a 4-dimensional compact minimal contact CR-warped product submanifold is homeomorphic to a sphere if ∣∇lnf∣2 < 1 is satisfied. By using Bonnet-Myers’s theorem we give a result about fundamental group and by Leung’s theorem we obtain a result about homology groups of a contact CR-warped submanifold.

Sphere theorems with and without smoothing

We show two sphere theorems for the Riemannian manifolds with scalar curvature bounded below and the non-collapsed mathrm {RCD}(n-1,n) spaces with mean distance close to frac{pi }{2}.

A quasiconformal Hopf soap bubble theorem

We show that any compact surface of genus zero in {mathbb {R}}^3 that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old open problem by H. Hopf, and gives a spherical version of Simon’s quasiconformal Bernstein theorem. The result generalizes, among others, Hopf’s theorem for constant mean curvature spheres, the classification of round spheres as the only compact elliptic Weingarten surfaces of genus zero, and the uniqueness theorem for ovaloids by Han, Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of solutions to linear elliptic equations with discontinuous coefficients.

Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold $M$ to a connected compact Riemannian manifold $N$, where $\dim M \geq \dim N$, has no singular points on $M$ in the sense of Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb sphere theorem for Lipschitz functions, i.e., if a closed Riemannian manifold admits a Lipschitz function with exactly two singular points in the sense of Clarke, then the manifold is homeomorphic to the sphere.

Volume pinching theorems for CAT(1) spaces

We examine volume pinching problems of ${\rm CAT}(1)$ spaces. We characterize a class of compact geodesically complete ${\rm CAT}(1)$ spaces of small specific volume. We prove a sphere theorem for compact ${\rm CAT}(1)$ homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume growths under an upper curvature bound.

Isometric immersions of RCD spaces

We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian manifolds.

Ricci flow and a sphere theorem for Ln/2-pinched Yamabe metrics

We obtain a differential sphere and Ricci flow convergence theorem for positive scalar curvature Yamabe metrics with Ln/2-pinched curvature in general dimensions n. Previously, E. Hebey and M. Vaugon obtained in [9] a corresponding result for Lp-pinching with p>n/2.

Extension theorems and a connection to the Erdős-Falconer distance problem over finite fields

The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide new L2→Lr extension estimates for paraboloids in certain odd dimensions with −1 non-square, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using the first association scheme graph to analyze energy sets, and as a consequence, we obtain new Lp→L4 extension theorems for spheres of primitive radii in odd dimensions, which break the Stein-Tomas result toward Lp→L4 which has stood for more than ten years. Most significantly, it follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the Lp→L4 estimates for spheres with primitive radii are much stronger than those for paraboloids. The second purpose is to show that there is a connection between the restriction conjecture associated to paraboloids and the Erdős-Falconer distance conjecture over finite fields. The last is to prove that the Erdős-Falconer distance conjecture holds in odd dimensional spaces when we study distances between two sets: one set lies on a variety (a paraboloid or a sphere), and the other set is arbitrary in vector spaces over finite fields.

A spinorial proof of the rigidity of the Riemannian Schwarzschild manifold

We revisit and generalize a recent result of Cederbaum (Cederbaum in preparation; Cederbaum 2017 Oberwolfach Report) concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems (Bunting and Masood-ul-Alam 1987 Gen. Relativ. Gravit. 19 147; Gibbons, Ida and Shiromizu 2002 Phys. Rev. Lett. 89 041101; Hwang 1998 Geom. Dedicata 71 5) as well as the more recent uniqueness theorems for photon spheres (Cederbaum 2015 Contemp. Math. 667 86; Cederbaum and Galloway 2017 Commun. Anal. Geom. 25 303; Cederbaum and Fehrenbach personal communication).

The Lawson-Simons' theorem on warped product submanifolds with geometric information

<abstract> The main objective of this paper is to investigate topological properties from the view point of compact warped product submanifolds of a space form with the vanishing constant sectional curvature. That is, we prove the non-existence of stable integral $ p $-currents in a compact oriented warped product pointwise semi-slant submanifold $ M^n $ in the Euclidean space $ \mathbb{R}^{p+2q} $ which satisfies an operative condition involving the Laplacian of a warped function and a pointwise slant function, and show that their homology groups are zero on this operative condition. Moreover, under the assumption of extrinsic conditions, we derive new topological sphere theorems on a warped product submanifold $ M^n $, and prove that $ M^n $ is homeomorphic to $ \mathbb{S}^n $ if $ n = 4 $, and $ M^n $ is homotopic to $ \mathbb{S}^n $ if $ n = 3 $. Furthermore, the same results are generalized for CR-warped products and our results recovered ^{[<xref ref-type="bibr" rid="b17">17</xref>]}. </abstract>

Sphere theorems for Lagrangian and Legendrian submanifolds

In this paper, we prove some differentiable sphere theorems and topological sphere theorems for Lagrangian submanifolds in K\"ahler manifold and Legendrian submanifolds in Sasaki space form.