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.     

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.     

Q‐Curvature on a Class of Manifolds with Dimension at Least 5

For a smooth compact Riemannian manifold with positive Yamabe invariant, positive Q‐curvature, and dimension at least 5, we prove the existence of a conformal metric with constant Q‐curvature. Our approach is based on the study of an extremal problem for a new functional involving the Paneitz operator.© 2016 Wiley Periodicals, Inc.     

Rigidité conforme des hémisphères {{\rm S}^4_+} et {{\rm S}^6_+}

Let (M, g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its Yamabe invariant is positive, then (M, g) is conformally isometric to the standard hemisphere. As an application and using a result of Hang and Wang (Comm Anal Geom 14(1):91–106, 2006), we prove a rigidity result for these hemispheres regarding the Min-Oo conjecture.     

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.     

Yamabe 流下完备Riemannian 流形在无穷远的性质

本文研究完备的局部共形平坦的Riemannian 流形Mⁿ. 证明了在Yamabe 流下, 流形在无穷远处曲率趋向于零的性质是随时间保持的. 作为应用, 可以得到这个流形的渐近体积比是一个常数.     

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 sphere theorem on locally conformally flat even-dimensional manifolds

In this paper, we prove that a closed even-dimensional manifold which is locally conformally flat with positive scalar curvature, positive Euler characteristic and which satisfies some additional condition on its curvature is diffeomorphic to the sphere or projective space.     

Transformations of locally conformally Kähler manifolds

We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal.     

Compactness for conformal metrics with constant Q curvature on locally conformally flat manifolds

In this note we study the conformal metrics of constant Q curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension n ≥ 5 and with Poincaré exponent less than , the set of conformal metrics of positive constant Q and positive scalar curvature is compact in the C∞ topology.     

The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank

In this paper we show that the geodesic flow on a compact locally symmetric space of nonpositive curvature has a unique invariant measure of maximal entropy. As an application to dynamics we show that closed geodesics are uniformly distributed with respect to this measure. Furthermore, we prove that the volume entropy is minimized at a compact locally symmetric space of nonpositive curvature among all conformally equivalent metrics with the same total volume.     

Theorems of existence and of vanishing of conformally killing forms

On an n-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric 2-tensors. We prove that, if the curvature operator is negative, then the manifold admits no nonzero conformally Killing p-forms for p = 1, 2, …, n ? 1. On the other hand, we prove that the dimension of the vector space of conformally Killing p-forms on an n-dimensional compact simply-connected conformally flat Riemannian manifold (M,g) is not zero.     

具有调和曲率与有限$L^p$曲率的完备流形

Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.     

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.     

Q-curvature flow on 4-manifolds with boundary

Given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero. Using integral methods, we prove global existence and convergence for the Q-curvature flow to a smooth metric conformal to g of prescribed Q-curvature, zero T-curvature and vanishing mean curvature under conformally invariant assumptions.     

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.     

Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds

We study compact complex manifolds bimeromorphic to locally conformally Kähler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kähler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim \(\ge 1\)) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds.     

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.     

Yamabe Invariants and $ Spin^c $ Structures

The Yamabe invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive but strictly less than that of the 4-sphere. This is done by using Dirac operators to control the lowest eigenvalue of a perturbation of the Yamabe Laplacian. These results dovetail perfectly with those derived from the perturbed Seiberg–Witten equations [Le3], but the present method is much more elementary in spirit. Submitted: October 1997     

Positive mass theorem for the Yamabe problem on spin manifolds

Let (M, g) be a compact connected spin manifold of dimension n ≥ 3 whose Yamabe invariant is positive. We assume that (M, g) is locally conformally flat or that n ∈ {3, 4, 5}. According to a positive mass theorem by Schoen and Yau the constant term in the asymptotic development of the Green’s function of the conformal Laplacian is positive if (M, g) is not conformally equivalent to the sphere. The proof was simplified by Witten with the help of spinors. In our article we will give a proof which is even considerably shorter. Our proof is a modification of Witten’s argument, but no analysis on asymptotically flat spaces is needed.Received: March 2004 Revised: June 2004 Accepted: June 2004