RIGIDITY THEOREMS OF CLIFFORD TORUS

In this article, we prove that the Clifford torus S1(1-r2~(1/2)) × Sn-1(r) is the only closed hypersurface in the unit sphere Sn+1(1) with infinite fundamental group, which satisfy r2 ≥ (n-1)/n, RicM ≤ C-(H), and S ≤ S+(H). Moreover, we give a characterization of Clifford torus S1(1-r2~(1/2)) × Sn-1(r) with r2 = 2(n-1)+nH2±|H| n2H2+4(n-1)(1/2)/2n(1+H2) .     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.     

Conformal metrics and Ricci tensors on the sphere

We consider tensors on the unit sphere , where , is the standard metric and is a differentiable function on . For such tensors, we consider the problems of existence of a Riemannian metric , conformal to , such that , and the existence of such a metric that satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability, and we construct the solutions when they exist. We show that these metrics are unique up to homothety, and we characterize those defined on the whole sphere. As a consequence of these results, we determine the tensors that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor , 0 $">, has no solution on if and only metrics homothetic to admit as a Ricci tensor. We also show that if , then equation has no solution , conformal to on , and only metrics homothetic to are solutions to this equation when . Infinitely many solutions, globally defined on , are obtained for the equation

where . The geometric interpretation of these solutions is given in terms of existence of complete metrics, globally defined on and conformal to the Euclidean metric, for certain bounded scalar curvature functions that vanish at infinity.

     

On hypersurfaces with two distinct principal curvatures in a unit sphere

We investigate the immersed hypersurfaces in a unit sphere . By using Otsuki's idea, we obtain the local and global classification results for immersed hypersurfaces in of constant m-th mean curvature and two distinct principal curvatures of multiplicities n−1,1 (in the local version, we assume that the principal curvatures are non-zero when m2). As the result, we prove that any local hypersurface in of constant mean curvature and two distinct principal curvatures is an open part of a complete hypersurface of the same curvature properties. The corresponding result does not hold for m-th mean curvature when m2.     

On the four-dimensional T<Superscript>2</Superscript>-manifolds of positive Ricci curvature

We prove that there is a T ²-invariant Riemannian metric of positive Ricci curvature on every four-dimensional simply connected T ²-manifold.     

Some properties of the Schouten tensor and applications to conformal geometry

The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the th elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, . We prove that this eigenvalue condition for implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.

     

Sharp bounds for the first non-zero Stekloff eigenvalues

Let (M,,) be an n(2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problems

where Δ is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first non-zero eigenvalues of the above problems will be denoted by p₁ and q₁, respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c, then with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius , here λ₁ denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q₁nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius . Finally, we show that q₁A/V and that if the equality holds and if there is a point x₀∂M such that the mean curvature of ∂M at x₀ is no less than A/{nV}, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively.     

K?hler Manifolds with Almost Non-negative Ricci Curvature

Compact Kähler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kähler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} \), where X _j is a Calabi-Yau manifold, or a hyperKähler manifold, or X _j satisfies H ⁰(X _j , Ω ^p ) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kähler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ∈ > 0, there exists a Kähler structure (J _∈, g _∈) on M such that the volume \({\text{Vol}}_{{g_{ \in } }} {\left( M \right)} < V\), the sectional curvature |K(g _∈)| < Λ², and the Ricci-tensor Ric(g _∈)> ?∈g _∈, where V and Λ are two constants independent of ∈. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} \), where X _i is a Calabi-Yau manifold, or a hyperKähler manifold, or X _i satisfies H ⁰(X _i , Ω ^p ) = {0}, p > 0.     

Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton

The aim of this note has two folds. First, we show a gradient estimate of the higher eigenfunctions of the weighted Laplacian on smooth metric measure spaces. In the second part, we consider a gradient steady Ricci soliton and prove that there exists a positive constant c(n)$c(n)$ depending only on the dimension n   of the soliton such that there is no nontrivial harmonic 1-form (hence harmonic function) which is in L^p$L^p$ on such a soliton for any 2<p<c(n)$2<p<c(n)$.     

Smooth diameter and eigenvalue rigidity in positive Ricci curvature

A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given and there exists a positive constant 0$">such that any -dimensional complete Riemannian manifold with Ricci curvature , sectional curvature and diameter is Lipschitz close and diffeomorphic to the standard unit -sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.

     

A sphere theorem for odd-dimensional submanifolds of spheres

We establish a topological sphere theorem from the point of view of submanifold geometry for odd-dimensional submanifolds of a unit sphere. We give examples which show that our result is optimal. Moreover, we note the assumption that the dimension is odd is essential.

     

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then      

On Positive Sasakian Geometry

A Sasakian structure =(\xi,\eta,\Phi,g) on a manifold Mis called positiveif its basic first Chern class c₁( ) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This provides us with a new technique for proving the existence of positive Ricci curvature metrics on certain odd dimensional manifolds. As an example we give a completely independent proof of a result of Sha and Yang that for every nonnegative integer kthe 5-manifolds k#(S ²×S ³) admits metrics of positive Ricci curvature.     

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

Fixed point theorem for generalized Φ -pseudocontractive mappings

Let E$E$ be a real Banach space, C$C$ be a nonempty closed convex subset of E$E$ and T:C→C$T:C\to C$ be a continuous generalized Φ$\Phi$-pseudocontractive mapping. It is proved that T$T$ has a unique fixed point in C$C$.