Some properties of non-compact complete Riemannian manifolds

In this paper, we study the volume growth property of a non-compact complete Riemannian manifold M. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on M, for any q∈(0,∞), every non-negative Lq subharmonic function is constant under a natural decay condition on the Ricci curvature.     

Sobolev type inequalities on Riemannian manifolds

This paper studies Sobolev type inequalities on Riemannian manifolds. We show that on a complete non-compact Riemannian manifold the constant in the Gagliardo-Nirenberg inequality cannot be smaller than the optimal one on the Euclidean space of the same dimension. We also show that a complete non-compact manifold with asymptotically non-negative Ricci curvature admitting some Gagliardo-Nirenberg inequality is not very far from the Euclidean space.     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L~q-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an(n + k)-dimensional complete Riemannian manifold M of non-negative(n-1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L~q-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold (M) of non-negative (n-1)-th Ricci curvature.The Liouville type theorem about the p-harmonic map with finite Lq-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

Eigenvalue Monotonicity for the Ricci-Hamilton Flow

In this short note, we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We show that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. This result will be useful in the study of blow-up models of the Ricci-Hamilton flow. Mathematics Subject Classifications (1991): 53C44 In Memory of S.S. Chern     

Rigidity for Closed Totally Umbilical Hypersurfaces in Space Forms

In Perez (Thesis, 2011), Perez proved some L ² inequalities for closed convex hypersurfaces immersed in the Euclidean space ? ⁿ⁺¹, and more generally for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this paper, we discuss the rigidity of these inequalities when the ambient manifold is ? ⁿ⁺¹, the hyperbolic space ? ⁿ⁺¹, or the closed hemisphere \(\mathbb{S}_{+}^{n+1}\) . We also obtain a generalization of Perez’s theorem to the hypersurfaces without the hypothesis of non-negative Ricci curvature.     

Harmonic forms on manifolds with non-negative Bakry–Émery–Ricci curvature

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry–Émery–Ricci curvature if the space of weighted L ² harmonic one-forms is non-trivial, then the weighted volume of the manifold is finite and the universal cover of the manifold splits isometrically as the product of the real line with a hypersurface.     

p-harmonic 1-forms on complete manifolds

Let (M ^m , g) be a complete non-compact manifold with asymptotically non-negative Ricci curvature and finite first Betti number. We prove that any bounded set of p-harmonic 1-forms in L ^q (M), 0 < q < ∞, is relatively compact with respect to the uniform convergence topology.     

Bundle stabilisation and positive Ricci curvature

If E is the total space of a vector bundle over a compact Ricci non-negative manifold, it is known that E×Rp admits a complete metric of positive Ricci curvature for all sufficiently large p. In this paper we establish a small, explicit lower bound for the dimension p.     

Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p∈M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff convergence theory, that, if model volume growth is sufficiently close to 1, then M is diffeomorphic to Euclidean n-dimensional space. Hence, our main theorem has various advantages of the Cheeger-Colding diffeomorphism theorem via the Euclidean volume growth. Our main theorem also contains a result of do Carmo and Changyu as a special case.     

L 2 Forms and Ricci Flow with Bounded Curvature on Complete Non-Compact Manifolds

In this paper, we study the evolution of L ² one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the L ² norm of a smooth one form is non-increasing along the Ricci flow with bounded curvature. The L ^∞ norm is showed to have monotonicity property too. Then we use L ^∞ cohomology of one forms with compact support to study the singularity model for the Ricci flow on .     

Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.     

Remarks on scalar curvature of Yamabe solitons

In this article, we consider the scalar curvature of Yamabe solitons. In particular, we show that, with natural conditions and non-positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that a complete non-compact Yamabe soliton with the quadratic decay at infinity of its Ricci curvature has non-negative scalar curvature. A new proof of Kazdan?CWarner condition is also presented.     

Non-compact $$\text {RCD}(0,N)$$ Spaces with Linear Volume Growth

Since non-compact \(\text {RCD}(0, N)\) spaces have at least linear volume growth, we study non-compact \(\text {RCD}(0,N)\) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grows at most linearly on a non-compact \(\text {RCD}(0,N)\) space satisfying the linear volume growth condition. Another main result in this paper is a rigidity theorem at the non-compact end for a \(\text {RCD}(0,N)\) space with strongly minimal volume growth. These results generalize some theorems on non-compact manifolds with non-negative Ricci curvature to non-smooth settings.     

The Gagliardo-Nirenberg inequalities and manifolds of non-negative Ricci curvature

This article is devoted to show that complete non-compact Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to two in which some Gagliardo-Nirenberg type inequality holds are not very far from the Euclidean space.     

Generalized quasi-Einstein manifolds with harmonic Weyl tensor

In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n ? 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.     

On complete manifolds supporting a weighted Sobolev type inequality

This paper studies the geometric and topological properties of complete open Riemannian manifolds which support a weighted Sobolev or log-Sobolev inequality. We show that the constant in the weighted Sobolev inequality on a complete open Riemannian manifold should be bigger than or equal to the optimal one on the Euclidean space of the same dimension and that a complete open manifold of asymptotically non-negative Ricci curvature supporting a weighted Sobolev inequality must have large volume growth. We also show that a complete manifold of non-negative Ricci curvature on which the log-Sobolev inequality holds is not very far from the Euclidean space.     

“A priori” estimates, uniqueness and existence of positive solutions of Yamabe type equations on complete manifolds

We study the asymptotic behaviour of non-negative solutions of Yamabe type equations on a complete Riemannian manifold. Then we provide a comparison result, based on a form of the weak maximum principle at infinity, which together with the “a priori” estimates previously obtained, yields uniqueness under very general Ricci assumptions. The paper ends with an existence result and an application to the non-compact Yamabe problem.     

On the conditions to control curvature tensors of Ricci flow

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M ⁿ , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that L^\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C ⁰ bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M ⁿ  × [0, T), the curvature tensor stays uniformly bounded on M ⁿ  × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.     

Almost Schur lemma for manifolds with boundary

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.