Sphere Theorem Research Articles

Differentiable sphere theorems whose comparison spaces are standard spheres or exotic ones

We show that for an arbitrarily given closed Riemannian manifold $M$ admitting a point $p \in M$ with a single cut point, every closed Riemannian manifold $N$ admitting a point $q \in N$ with a single cut point is diffeomorphic to $M$ if the radial curvatures of $N$ at $q$ are sufficiently close in the sense of $L^1$-norm to those of $M$ at $p$.

Sphere theorems for RCD and stratified spaces

We prove topological sphere theorems for RCD(n-1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.

Sphere theorems for submanifolds in Kähler manifold

Sphere theorems for submanifolds in Kähler manifold

Stable currents and homology groups in a compact CR-warped product submanifold with negative constant sectional curvature

The main purpose of this paper is to highlight some striking features of the relationship between homology groups and compact warped product submanifolds geometry with negative constant sectional curvature that follows from Lawson and Simons (1973). Using the result of Fu and Xu (2008), we prove that there does not exist stable integral p-currents and their homology groups are zero in a compact CR-warped product submanifold Mn from a complex hyperbolic space ℂHm(−4) under extrinsic conditions involving the Laplacian of warped function and the squared norm of gradient of the warping function. Moreover, under the same extrinsic conditions and using the generalized Poincaré’s conjecture, we derive new topological sphere theorem on a compact CR-warped product submanifold Mn, we prove that Mn is homeomorphic to the sphere Sn if n=4. Also, if n=3 then M3 is homotopic to the sphere S3 and this result follows from the work of Sjerve (1973). Finally, the same types of theorems are constructed in terms of some mechanical tools such as Dirichlet energy and Hamiltonian as well.

Quantitative volume space form rigidity under lower Ricci curvature bound I

Let $M$ be a compact $n$-manifold of $\mathrm{Ric}_M \geq (n - 1) H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $ \rho \gt 0$ such that for any $x \in M$, the open $ \rho $-ball at $x^{\ast}$ in the (local) Riemannian universal covering space, $ (U^{\ast}_{\rho} , x^{\ast}) \to (B_{\rho} (x) , x)$, has the maximal volume, i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H = -1$, the volume entropy of $M$ is maximal, i.e., $n - 1$ ([LW1]). The main results of this paper are quantitative space form rigidity, i.e., statements that $M$ is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H = 1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].

Ideal Hyperbolic Polyhedra and Discrete Uniformization

We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichmuller spaces $$\widetilde{\mathscr {T}}_{g,n}$$ of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over $$\mathscr {T}_{g,n}$$ , and invariant under the action of the mapping class group.

Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption

Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry’s sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $$\mathop {\mathrm {Ric}}\nolimits \ge -Kg$$ and $$\mathop {\mathrm {diam}}\nolimits \le D$$ for some positive constants $$K>0$$ and $$D>0$$ .

Some sphere theorems in linear potential theory

In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, we prove that if the mean curvature $H$ of the boundary obeys the condition $$ - \bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} \leq \frac{H}{n-1} \leq \bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} , $$ then $\Omega$ is a round ball.

A generalized maximal diameter sphere theorem

We prove that if a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\widetilde M$ belonging to a certain class, then the diameter of $M$ does not exceed that of $\widetilde M$. Moreover, we prove that if the diameter of $M$ equals that of $\widetilde M$, then $M$ is isometric to the $n$-model of $\widetilde M$. The class of a two-sphere of revolution employed in our main theorem is very wide. For example, this class contains both ellipsoids of prolate type and spheres of constant sectional curvature. Thus our theorem contains both the maximal diameter sphere theorem proved by Toponogov [9] and the radial curvature version by the present author [2] as a corollary.

A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

A classic gap theorem for complete non-compact Ricci flat manifolds by M. Anderson suggests that one can get the rigidity of certain spaces simply by passing local curvature estimates on geodesic balls to global. Using similar ideas, Kim showed that a 4-dimensional complete non-compact Bach-flat manifold with vanishing scalar curvature and small L2-curvature tensor has to be flat. Unfortunately, this method does not generalize to compact manifolds without boundary. Applying a different approach, we show that a closed Bach-flat Riemannian manifold with positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either L∞ or Ln2-norm. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A. Singer. As an application, we can partially recover the well-known Chang–Gursky–Yang's 4-dimensional conformal sphere theorem.

Some differentiable sphere theorems

In this paper, we obtain several new intrinsic and extrinsic differentiable sphere theorems via Ricci flow. For intrinsic case, we show that a closed simply connected $$n(\ge 4)$$ -dimensional Riemannian manifold M is diffeomorphic to $$\mathbb {S}^n$$ if one of the following conditions holds pointwisely: $$\begin{aligned} (i)\ R_0>\left( 1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right) K_{max};\quad \ (ii)\ \frac{Ric^{[4]}}{4(n-1)}>\left( 1-\frac{6(\sqrt{10}-3)}{n-1}\right) K_{max}. \end{aligned}$$ Here $$K_{max}$$ , $$Ric^{[k]}$$ and $$R_0$$ stand for the maximal sectional curvature, the k-th weak Ricci curvature and the normalized scalar curvature. For extrinsic case, i.e., when M is a closed simply connected $$n(\ge 4)$$ -dimensional submanifold immersed in $$\bar{M}$$ . We prove that M is diffeomorphic to $$\mathbb {S}^n$$ if it satisfies some curvature pinching conditions. The only involved extrinsic quantities in our pinching conditions are the maximal sectional curvature $$\bar{K}_{max}$$ and the squared norm of mean curvature vector $$\left|H\right|^2$$ . More precisely, we show that M is diffeomorphic to $$\mathbb {S}^n$$ if one of the following conditions holds: It is worth pointing out that, in the proof of extrinsic case, we apply suitable complex orthonormal frame and simplify the calculations considerably. We also emphasize that both of the pinching constants in (2) and (3) are optimal for $$n=4$$ .

Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4‐manifolds

AbstractThe goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4‐manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.

Revisiting the Birkhoff theorem from a dual null point of view

The Birkhoff theorem is a well-known result in general relativity and it is used in many applications. However, its most general version, due to Bona, is almost unknown and presented in a form less accessible to the relativist and cosmologist community. Moreover, many wield it mistakenly as a simple transposition of Newton's iron sphere theorem. In the present work, we propose a modern, dual null, presentation --- useful in many explorations, including black holes --- of the theorem that renders accessible most of the results of Bona's version. In addition, we discuss the fluid contents admissible for the application of the theorem, beyond a vacuum, and we demonstrate how the formalism greatly simplifies solving the dynamical equations and allows one to express the solution as a power expansion in $r$. We present a family of solutions that share the properties predicted by the Birkhoff theorem and discuss the existence of trapped and antitrapped regions. The formalism manifestly shows how the type of region --- trapped or untrapped --- determines the character of the Killing vector.

Four-manifolds with positive Yamabe constant

We refine Theorem A due to Gursky \cite{G3}. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality. We prove some classification theorems of four manifolds according to some conformal invariants (see Theorems 1.3 and 1.6), which generalize the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}.

The Weighted Connection and Sectional Curvature for Manifolds With Density

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion-free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger’s finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.

Strictly positive definite multivariate covariance functions on spheres

We study the strict positive definiteness of matrix-valued covariance functions associated to multivariate random fields defined over d-dimensional spheres of the (d+1)-dimensional Euclidean space. Characterization of strict positive definiteness is crucial to both estimation and cokriging prediction in classical geostatistical routines. We provide characterization theorems for high dimensional spheres as well as for the Hilbert sphere. We offer a necessary condition for positive definiteness on the circle. Finally, we discuss a parametric example which might turn to be useful for geostatistical applications.

Overdetermined elliptic problems in topological disks

We introduce a method, based on the Poincaré–Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.

The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

On a compact quaternionic contact (qc) manifold of dimension bigger than seven and satisfying a Lichnerowicz type lower bound estimate we show that if the rst positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard 3-Sasakian sphere. The same conclusion is shown to hold on a non-compact qc manifold which is complete with respect to the associated Riemannian metric assuming the existence of a function with traceless horizontal Hessian.

Generalized packing radius theorems of Alexandrov spaces with curvature ≥ 1

In this paper, we give some generalized packing radius theorems of an [Formula: see text]-dimensional Alexandrov space [Formula: see text] with curvature [Formula: see text]. Let [Formula: see text] be any [Formula: see text]-separated subset in [Formula: see text] (i.e. the distance [Formula: see text] for any [Formula: see text]). Under the condition “[Formula: see text]” (after [K. Grove and F. Wilhelm, Hard and soft packing radius theorems, Ann. of Math. 142 (1995) 213–237]), we give the upper bound of [Formula: see text] (which depends only on [Formula: see text]), and classify the geometric structure of [Formula: see text] when [Formula: see text] attains the upper bound. As a corollary, we get an isometrical sphere theorem in Riemannian case.

Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold M into a Riemannian manifold N admits a smooth approximation via immersions if the map has no singular points on M in the sense of F.H.Clarke, where dimM≤dimN. As its corollary, we have that if a bi-Lipschitz homeomorphism between compact manifolds and its inverse map have no singular points in the same sense, then they are diffeomorphic. We have three applications of the main theorem: The first two of them are two differentiable sphere theorems for a pair of topological spheres including that of exotic ones. The third one is that a compact n-manifold M is a twisted sphere and there exists a bi-Lipschitz homeomorphism between M and the unit n-sphere Sn(1) which is a diffeomorphism except for a single point, if M satisfies certain two conditions with respect to critical points of its distance function in the Clarke sense. Moreover, we have three corollaries from the third theorem; the first one is that for any twisted sphere Σn of general dimension n, there exists a bi-Lipschitz homeomorphism between Σn and Sn(1) which is a diffeomorphism except for a single point. In particular, there exists such a map between an exotic n-sphere Σn of dimension n>4 and Sn(1); the second one is that if an exotic 4-sphere Σ4 exists, then Σ4 does not satisfy one of the two conditions above; the third one is that for any Grove–Shiohama type n-sphere N, there exists a bi-Lipschitz homeomorphism between N and Sn(1) which is a diffeomorphism except for one of points that attain their diameters.