Nonnegative Ricci Curvature Research Articles

The growth rate of harmonic functions

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than three, we prove that quantitative strong unique continuation yields the existence of nonconstant polynomial growth harmonic functions. Also the uniform bound of frequency for linear growth harmonic functions on such manifolds is obtained, and this confirms a special case of Colding-Minicozzi conjecture on frequency.

Monotonicity and rigidity of the $${\mathcal {W}}$$-entropy on $${\mathsf {RCD}} (0,N)$$ spaces

By means of a space-time Wasserstein control, we show the monotonicity of the $${\mathcal {W}}$$ -entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipation of the $${\mathcal {W}}$$ -entropy is also proved. These extend known results even on weighted Riemannian manifolds in some respects. In addition, we reveal that some singular spaces will exhibit the rigidity models while only the Euclidean space does in the class of smooth weighted Riemannian manifolds.

Superlinear elliptic inequalities on manifolds

Let M be a complete non-compact Riemannian manifold and let σ be a Radon measure on M. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequality−Δu≥σuqinM, where q>1. We obtain necessary and sufficient criteria for existence of positive solutions in terms of Green function of Δ. In particular, explicit necessary and sufficient conditions are given when M has nonnegative Ricci curvature everywhere in M, or more generally when Green's function satisfies the 3G-inequality.

Cahen–Gutt moment map, closed Fedosov star product and structure of the automorphism group

We show that if a compact Kaehler manifold $M$ of non-negative Ricci curvature admits closed Fedosov star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente-Gravy. More generally we consider the squared norm of Cahen-Gutt moment map as in the same spirit of Calabi functional for the scalar curvature in cscK problem, and prove a Cahen-Gutt version of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kaehler manifolds. The proof uses a Hessian formula for the squared norm of Cahen-Gutt moment map.

Gradient estimate of positive eigenfunctions of sub-Laplacian on complete pseudo-Hermitian manifolds

In this paper, we derive an explicit gradient estimate of positive solutions of Δbu=−λu on complete noncompact pseudo-Hermitian manifolds with bounded geometric conditions. As a consequence, we obtain an estimate of the greatest lower bound for the L2-spectrum of the sub-Laplacian operator. Moreover, we recapture the Liouville theorem of positive pseudo-harmonic functions on complete noncompact Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.

Nonnegative Ricci curvature, stability at infinity and finite generation of fundamental groups

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose maximal Euclidean factor has dimension $k$, then $\pi_1(M)$ is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and the unique tangent cone at infinity.

Energy-minimizing maps from manifolds with nonnegative Ricci curvature

The energy of any [Formula: see text] representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only if the original map is totally geodesic. This conclusion also holds under the weaker assumption that the domain is finitely covered by a diffeomorphic product, and its universal covering space splits isometrically as a product with a flat factor, in a commutative diagram that follows from the Cheeger–Gromoll splitting theorem.

The concavity of p-Rényi entropy power for doubly nonlinear diffusion equations and Lp-Gagliardo-Nirenberg-Sobolev inequalities

We prove the concavity of p-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on Rn or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp Lp-Sobolev inequality and Lp-Gagliardo-Nirenberg inequalities on Rn. Moreover, two improvements of Lp-Gagliardo-Nirenberg inequalities are derived.

A characterization related to Schrödinger equations on Riemannian manifolds

In this paper, we consider the following problem: [Formula: see text] where [Formula: see text] is a [Formula: see text]-dimensional ([Formula: see text], non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, [Formula: see text] is a real parameter, [Formula: see text] is a positive coercive potential, [Formula: see text] is a bounded function and [Formula: see text] is a suitable nonlinearity. By using variational methods, we prove a characterization result for existence of solutions for [Formula: see text].

On the Dimensions of Spaces of Harmonic Functions with Polynomial Growth

In this paper, we obtain an estimate for the lower bound for the dimensions of harmonic functions with polynomial growth and a Liouville type theorem on manifolds with nonnegative Ricci curvature whose tangent cone at infinity is a unique metric cone with a conic measure.

A Bochner formula for harmonic maps into non-positively curved metric spaces

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(\(-\,1\)) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eells–Sampson Bochner formula in this more general setting. In particular, we show that harmonic maps from spaces of non-negative Ricci curvature into non-positively curved spaces have subharmonic energy density. When the domain is compact the energy density is constant, and if the domain has a point of positive Ricci curvature every harmonic map into an NPC space must be constant.

Some regularity results for [formula omitted]-harmonic mappings between Riemannian manifolds

Let M be a C2-smooth Riemannian manifold with boundary and N a complete C2-smooth Riemannian manifold. We show that each stationary p-harmonic mapping u:M→N, whose image lies in a compact subset of N, is locally C1,α for some α∈(0,1), provided that N is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing p-harmonic mappings with image being contained in a regular geodesic ball. Moreover, when M has non-negative Ricci curvature and N is simply connected with non-positive sectional curvature, we deduce a gradient estimate for C1-smooth weakly p-harmonic mappings from which follows a Liouville-type theorem in the same setting.

On the fundamental group of complete manifolds with almost Euclidean volume growth

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.

Local estimate of fundamental groups

For any complete n-dim Riemannian manifold Mn with nonnegative Ricci curvature, Kapovitch and Wilking proved that any finitely generated subgroup of the fundamental group π1(Mn) can be generated by C(n) generators. Inspired by their work, we give a quantitative proof of the above theorem and show that C(n)≤nn20n. Our main tools are quantitative Cheeger-Colding's almost splitting theory, and the squeeze lemma for covering groups between two Riemannian manifolds with nonnegative Ricci curvature.

On the three-circle theorem and its applications in Sasakian manifolds

This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of K\"ahler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau's uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward second and third CR Yau's uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.

Pseudo-harmonic Maps from Complete Noncompact Pseudo-Hermitian Manifolds to Regular Balls

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem is established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to regular balls of Riemannian manifolds.

Gap theorem on Kähler manifolds with nonnegative orthogonal bisectional curvature

Abstract In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam, Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, Duke Math. J. 165 2016, 15, 2899–2919].

The concavity of Rényi entropy power for the parabolic p-Laplace equations and applications

In this paper, we prove that the concavity of Renyi entropy power of positive solutions to the parabolic p-Laplace equations on compact Riemannian manifold with nonnegative Ricci curvature. As applications, we derive the improved $$L^p$$ -Gagliardo-Nirenberg inequalities.

Integral of Scalar Curvature on Non-parabolic Manifolds

Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold \((M^3, g)\) with non-negative Ricci curvature, the asymptotic- weighted scaling invariant integral of scalar curvature has an explicit bound in the form of asymptotic volume ratio.

Totally geodesic maps into manifolds with no focal points

The set of totally geodesic representatives of a homotopy class of maps from a compact Riemannian manifold $M$ with nonnegative Ricci curvature into a complete Riemannian manifold $N$ with no focal points is path-connected and, when nonempty, equal to the set of energy-minimizing maps in that class. When $N$ is compact, each map from a product $W \times M$ into $N$ is homotopic to a map that's totally geodesic on each $M$-fiber. These results may be used to extend to the case of no focal points a number of splitting theorems of Cao-Cheeger-Rong about manifolds with nonpositive sectional curvature and, in turn, to generalize a non-collapsing theorem of Heintze-Margulis. In contrast with previous approaches, they are proved using neither a geometric flow nor the Bochner identity for harmonic maps.