We prove that every closed, universally embeddable CR three-manifold with nonnegative Yamabe constant and positive total Q ′ Q^\prime -curvature is contact diffeomorphic to a quotient of the standard contact three-sphere. We also prove that every closed, embeddable CR three-manifold with zero Yamabe constant and nonnegative total Q ′ Q^\prime -curvature is CR equivalent to a compact quotient of the Heisenberg group with its flat CR structure.

Given an uncountable subset Y \mathcal {Y} of a nonseparable Banach space, is there an uncountable Z ⊆ Y \mathcal {Z}\subseteq \mathcal {Y} such that the distances between any two distinct points of Z \mathcal {Z} are more or less the same? If an uncountable subset Y \mathcal {Y} of a nonseparable Banach space does not admit an uncountable Z ⊆ Y \mathcal {Z}\subseteq \mathcal {Y} , where any two points are distant by more than r > 0 r>0 , is it because Y \mathcal {Y} is the countable union of sets of diameters not bigger than r r ? Clearly, these types of questions can be rephrased in the combinatorial language of partitions of pairs of points of a Banach space X \mathcal {X} induced by the distance function d : [ X ] 2 → R + d:[\mathcal {X}]^2\rightarrow \mathbb {R}_+ . We investigate connections between the set-theoretic phenomena involved and the geometric properties of uncountable subsets of nonseparable Banach spaces of densities up to 2 ω 2^\omega related to uncountable ( 1 + ) (1+) -separated sets, equilateral sets or Auerbach systems. The results include geometric dichotomies for a wide range of classes of Banach spaces, some in ZFC, some under the assumption of OCA + + MA and some under a hypothesis on the descriptive complexity of the space as well as constructions (in ZFC or under CH) of Banach spaces where the geometry of the unit sphere displays anti-Ramsey properties. This complements classical theorems for separable spheres and the recent results of Hájek, Kania, Russo for densities above 2 ω 2^\omega as well as offers a synthesis of possible phenomena and categorization of examples for uncountable densities up to 2 ω 2^\omega obtained previously by the author and Guzmán, Hrušák, Ryduchowski and Wark.

A problem that geometers have always been concerned with is when a closed manifold is isometric to a round sphere. A classical result shows that a closed locally conformally flat Einstein manifold is always isometric to a quotient of a round sphere. In this note, we provide the definitions of σk-curvatures and σk-Einstein manifolds, and we show that a closed σk-Einstein manifold under certain pinching conditions of a Weyl curvature and Einstein curvature is isometric to a quotient of a round sphere.

One of the most important results in differential geometry is that the only closed hypersurfaces of constant mean curvature and in general constant higher order mean curvature) embedded in Euclidean space are round spheres [1]. This result is not true for the case of immersed (and non-embedded hypersurfaces [11, 14]. Many generalizations of this result have been obtained later, for example constant scalar curvatures or constant higher order mean curvatures hypersurfaces [2,3,7,9]. As a natural generalization of hypersurfaces with constant mean curvature or with constant higher order mean curvature, linear Weingarten and more general generalized, Weingarten hypersurfaces hypersurface has been studied in many places. [5],[10]. The aim of our work is to establish a characterization theorem concerning complete generalized Weingarten hypersurfaces embedded in Euclidean space. That is an hypersurfaces where some of the higher order mean curvature are linearly related. We prove that the only closed, oriented generalized Weingarten hypersurfaces embedded in Euclidean space with non-vanishing higher order mean curvature are the round spheres. This result generalizes the cases of constant higher order mean curvature hypersurfaces and linear Weingarten hypersurfaces embedded in Euclidean space.

In the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms and ordinary Lichnerowicz Laplacian acting on symmetric tensors. Additionally, we prove vanishing theorems for the null spaces of these Laplacians and find estimates for their lowest eigenvalues on closed Riemannian manifolds with suitably bounded curvature operators of the first kind, sectional and Ricci curvatures. Specifically, we will prove our version of the famous differential sphere theorem, which we will apply to the aforementioned problems concerning the ordinary Lichnerowicz Laplacian.

We prove Strichartz estimates for the semigroups associated to stratified and/or rotating inviscid geophysical fluids using Fourier restriction theory. We prove new results for rotating stratified fluids, and recover results from Koh, Lee, Takada, 2014 for rotation only, and from Lee, Takada, 2017 for stratification only. Our restriction estimates are obtained by the slicing method (Nicola 2009), which relies on the well-known Tomas–Stein theorem for 2-dimensional spheres. To our knowledge, such a method has never been used in this setting. Moreover, when the fluid is stratified, our approach yields sharp estimates, showing that the slicing method captures all the available curvature of the surfaces of interest.

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.

abstract: In this note, we study the Gehring link problem in the round sphere, which motivates our study of the width of a band in positively curved manifolds. Using the same idea, we are able to prove a sphere theorem for hypersurface in the round $\SS^n$ provided that its normal injectivity radius is large. A rigidity theorem for Clifford hypersurfaces in $\SS^n$ is also proved. The $3$-dimensional case of our theorems confirm two conjectures raised by Gromov (2018).

In this work, we study a functional involving the generalized scalar curvatures and prove a spherical sphere theorem under some pinching condition of this quantity. As an application, we define a new invariant on 3-dimensional manifolds and use it to study the topology of manifolds.

In this paper, taking into account that the 6-dimensional unit sphere is nearly Kaehler manifold, 4-dimensional CR-warped product submanifolds of the sphere are studied. First, an interesting relation is obtained among the warping function of the CR-warped product submanifold, the scalar curvature of the fibers and the components of the second fundamental form. Using this relation, topological and differential sphere theorems are given and totally geodesicity of CR-warped product submanifold of the 6-dimensional sphere is obtained. Moreover, a result is presented about homology groups of a CR-warped submanifold.

We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen’s g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field ℚ. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.

The geometry of [formula omitted]-harmonic maps

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 $${\textbf{S}}_{c}^3(\lambda )$$ .

Abstract In 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.

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

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.

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.

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.

Abstract In 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.