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This paper focuses on the study of the prescribed mean curvature problem on the unit ball. If the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small, then the existence of positive solutions of conformal mean curvature equation has been known. The purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method.

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo’s conjecture such as that of Brendle–Marques–Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.

We investigate the finiteness structure of a complete non-compact $n$-dimensional Riemannian manifold $M$ whose radial curvature at a base point of $M$ is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than $\pi$. We show, as our main theorem, that all Busemann functions on $M$ are exhaustions, and that there exists a compact subset of $M$ such that the compact set contains all critical points for any Busemann function on $M$. As corollaries by the main theorem, $M$ has finite topological type, and the isometry group of $M$ is compact., Comment: This version 2 is a version to appear in Math. Ann. 16 pages, No figures

We consider a C1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed p-mean curvature \({H\in C^{0},}\) we show that any characteristic curve is C2 smooth and its (line) curvature equals − H in the nonsingular domain. By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are C2 smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. These results can be applied to more general situations.

We derive a Harnack type inequality for the conformal scalar curvature equation \(\Delta u+K(x)u^\frac{n+2}{n-2}=0\) on B3R. If the positive scalar curvature function K(x) is sub-harmonic in a neighborhood of each critical point and the maximum of u over BR is comparable to its maximum over B3R, then the Harnack type inequality can be obtained.

For a Riemannian closed spin manifold and under some topological assumption (non-zero $\hat{A}$-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the first eigenvalue of the Laplacian. The main difficulty lies in the study of the odd-dimensional case. On the other hand, we study the equality case for the closed spin Riemannian manifolds with non-zero $\hat{A}$-genus. This work improves an inequality which was first proved by K. Ono in 1988., Comment: 29 pages

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric., Comment: 25 pages

By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature, which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary and sufficient condition under which the main existence results in [Ni1] and [KN] on prescribing nonnegative scalar curvature will hold. This condition had been sought in several papers in the last two decades.

Scalar curvature is the simplest generalization of Gaussian curvature to higher dimensions. However there are many questions open with regard to its relation to other geometric quantities and topology. Here we will prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curvature, namely the one-sided affinity for curvature decreasing deformations. The first one is concerned with some prescribed decrease of the scalar curvature Scal(g) of some Riemannian metric g on a given manifoldMn of dimension≥ 3. We denote the e−neighborhood of some set U with respect to g by Ue.

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Mathematics Subject Classification (1991): 53C20, 53C291. IntroductionLet M be a complete connected Riemannian manifold with boundary B, K bethe sectional curvature of the interior, and be the normal curvature of B. Theinradius of M isinr = supfd(p;B):p2Mg:Thus for every r

Let Rn+l(c) be an (n + 1)-dimensional Riemannian manifold with constant sectional curvature c, we also call it space form. When c -1, R"+l(c) = S "+l (i.e. (n + 1)-dimensional unit sphere space); when c = O, Rn+t(c) = E n+l (i.e. (n + 1)-dimensional Euclidean space). Let M be an n-dimensional compact hypersurface in Rn+l(c), and el . . . . . e,, a local orthonormal frame field on M, col . . . . . con its dual coframe field. Then the second fundamental form of M is

The problem of finding Riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of Kazdan and Warner in 1970's ([10] and references cited in it). In this paper we shall consider the same problem under a constraint on the volume. For this purpose it is useful to introduce an invariant p(M) of a smooth closed manifold M, which will be called the Yamabe number of M, defined as the supremum of #(M, C) of all conformal classes C of Riemannian metrics on M

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Har tman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of complete surfaces with constant curvature in R 3.

In studying the properties of the space of energy-finite solutions of the equation A u = Pu(P ~0) on an oriented Riemannian manifold R attention has been focused on the integral of P over neighborhoods of the Royden boundary points of R. In particular in [1] it was shown that the points of the Royden harmonic boundary that possess a neighborhood over which the integral of P is convergent completely determine this space. Recently Nakai [3] introduced the descriptive term P-nondensity point for points with this property. Off hand one would expect to be able to find a P that grows so rapidly that the P-nondensity points of the Royden boundary of R form a small set from the topological viewpoint. The purpose of this paper is to prove the rather startling fact (Theorem 2) that for any nonnegative locally integrable n-form P on a Riemannian n-manifold R the P-nondensity points on the Royden boundary of R form a dense subset. The key to this result is the compactification introduced by Nakai-Sario [4] and in Theorem 1 we give some relations between the Royden boundary and the boundary of this compactification. For a detailed account of the terminology and auxiliary results useA here see Chapter III of [51, [2] and [4]. We begin by fixing our notations. Let R be a Riemannian n-manifold, M its Royden algebra and M~ the potential subalgebra. The maximal ideal space R* of M with the weak* topology is the Royden compactification of of R. We denote by F = R*\R the Royden boundary and by d its harmonic part, i.e., A = {pER*lf(P)=O for every feM~}. Let P denote an arbitrary nonnegative locally integrable n-form on R and denote by E the subalgebra of M given by E = l f ~ M{~ f2p < + ~}. Under