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.     

Properties of complete non-compact Yamabe solitons

In this article, we first study the local volume estimate of the complete non-compact Yamabe soliton. Then we study the behavior of the potential function of the steady Yamabe soliton with positive Ricci curvature. We also study the scalar curvature decay of steady and expanding Yamabe solitons with Ricci pinching condition.     

The classification of locally conformally flat Yamabe solitons

This paper addresses the classification of locally conformally flat gradient Yamabe solitons. In the first part it is shown that locally conformally flat gradient Yamabe solitons with positive sectional curvature are rotationally symmetric. In the second part the classification of all radially symmetric gradient Yamabe solitons is given and their correspondence to smooth self-similar solutions of the fast diffusion equation on Rⁿ${\mathbb{R}}^n$ is shown. In the last section it is shown that any eternal solution to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space–time maximum must be a steady Yamabe soliton.     

Conformal entropy rigidity through Yamabe flows

We introduce two versions of the Yamabe flow which preserve negative scalar-curvature bounds. First we show existence and smooth convergence of solutions to these flows. We then show that a metric with negative scalar curvature is controlled by the Yamabe metrics in the same conformal class with constant extremal scalar curvatures. This implies that the volume entropy of our original metric is controlled by the entropies of these Yamabe metrics. We eventually use these Yamabe flows to prove an entropy-rigidity result: when the Yamabe metric has negative sectional curvature, the entropy of a metric in the same conformal class is extremal if and only if the metric has constant extremal scalar curvature.     

Complete gradient shrinking Ricci solitons have finite topological type

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:(i) the Ricci curvature is bounded from above;(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded. To cite this article: F.-q. Fang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension

Motivated by the definition of combinatorial scalar curvature given by Cooper and Rivin, we introduce a new combinatorial scalar curvature. Then we define the discrete quasi-Einstein metric, which is a combinatorial analogue of the constant scalar curvature metric in smooth case. We find that discrete quasi-Einstein metric is critical point of both the combinatorial Yamabe functional and the quadratic energy functional we defined on triangulated 3-manifolds. We introduce combinatorial curvature flows, including a new type of combinatorial Yamabe flow, to study the discrete quasi-Einstein metrics and prove that the flows produce solutions converging to discrete quasi-Einstein metrics if the initial normalized quadratic energy is small enough. As a corollary, we prove that nonsingular solution of the combinatorial Yamabe flow with nonpositive initial curvatures converges to discrete quasi-Einstein metric. The proof relies on a careful analysis of the discrete dual-Laplacian, which we interpret as the Jacobian matrix of curvature map.     

A note on complete manifolds with finite volume

In this article, we concern on complete manifolds with finite volume. We prove that under some assumptions about scalar curvature and the Yamabe constant, the manifolds must be compact, and we also give the diameter estimates in terms of the scalar curvature and the Yamabe constant.     

A NOTE ON COMPLETE MANIFOLDS WITH FINITE VOLUME

In this article, we concern on complete manifolds with finite volume. We prove that under some assumptions about scalar curvature and the Yamabe constant, the manifolds must be compact, and we also give the diameter estimates in terms of the scalar curvature and the Yamabe constant.     

The Yamabe problem for almost Hermitian manifolds

The conformal class of a Hermitian metric g on a compact almost complex manifold (M^2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature s^J. We ask if the conformal class of g carries a metric with constant s^J, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)s^J−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.     

On the Potential Function of Gradient Steady Ricci Solitons

In this paper, we study the potential function of gradient steady Ricci solitons. We prove that the infimum of the potential function decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential function must be trivial, and that no gradient steady Ricci soliton admits uniformly positive scalar curvature.     

The modified Yamabe problem and geometry of modified scalar curvatures

Study of modified scalar curvatures and the modified Yamabe problem is given. An existence theorem of type of Trudinger and Aubin is obtained. Interpreting Isotropic curvature as a certain modified scalar curvature we can extend the four-dimensional sphere theorem of [23]. Another modified scalar curvature is used in solving open problems concerning the conjecture of self-dual Einstein 4-manifold and the Goldberg conjecture, both being under negative Ricci condition.     

On bifurcation of solutions of the Yamabe problem in product manifolds

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds.     

On the $h$-Almost Yamabe Soliton

We introduce the concept $h$-almost Yamabe soliton which extends naturally the almost Yamabe soliton by Barbosa-Ribeiro and obtain some rigidity results concerning $h$-almost Yamabe solitons. Some condition for a compact $h$-almost Yamabe soliton to be a gradient soliton is also obtained. Finally, we give some characterizations for a special class of gradient $h$-almost Yamabe solitons.     

The exponential convergence of the CR Yamabe flow

In this paper, we study the CR(Cauchy-Riemann) Yamabe flow with zero CR Yamabe invariant.We use the CR Poincaré inequality and a Gagliardo-Nirenberg type interpolation inequality to show that this flow has the long time solution and the solution converges to a contact form with flat pseudo-Hermitian scalar curvature exponentially.     

Surgery and equivariant Yamabe invariant

We consider the equivariant Yamabe problem, i.e., the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their G-invariant conformal subclasses. We prove a formula about how the G-Yamabe invariant changes under the surgery of codimension 3 or more, and compute some G-Yamabe invariants.     

On perturbations of the fractional Yamabe problem

The fractional Yamabe problem, proposed by González and Qing (Analysis PDE 6:1535–1576, 2013), is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal conformally invariant operators. We investigate a non-compactness property of the fractional Yamabe problem by constructing bubbling solutions to its small perturbations.     

SCALAR CURVATURES ON NONCOMPACT RIEMANN MANIFOLDS

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First eigenvalues of geometric operators under the Yamabe flow

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.     

Surgery and the Yamabe Invariant

We study the Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular: for a compact connected manifold M with no metric of positive scalar curvature, we prove that the Yamabe invariant of any manifold obtained by performing surgery on spheres of codimension greater than 2 on M is not smaller than the invariant of M. Submitted: August 1998.     

A Construction of Constant Scalar Curvature Manifolds with Delaunay-type Ends

It has been showed by Byde (Indiana Univ. Math. J. 52(5):1147–1199, 2003) that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this paper is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to sufficiently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.