Annals of Global Analysis and Geometry最新文献

Given a closed Riemannian manifold M and (bge 2) closed connected submanifolds (N_jsubset M) of codimension at least 2, we prove that the first nonzero eigenvalue of the domain (Omega _varepsilon subset M) obtained by removing the tubular neighbourhood of size (varepsilon ) around each (N_j) tends to infinity as (varepsilon ) tends to 0. More precisely, we prove a lower bound in terms of (varepsilon ), b, the geometry of M and the codimensions and the volumes of the submanifolds and an upper bound in terms of (varepsilon ) and the codimensions of the submanifolds. For eigenvalues of index (k=b,b+1,ldots ), we have a stronger result: their order of divergence is (varepsilon ^{-1}) and their rate of divergence is only depending on m and on the codimensions of the submanifolds.

In this paper, we prove two rigidity results of hypersurfaces in n-dimensional weighted Riemannian manifolds with weighted scalar curvature bounded from below. Firstly, we establish a splitting theorem for the n-dimensional weighted Riemannian manifold via a weighted area-minimizing hypersurface. Secondly, we observe the topological invariance of the weighted stable hypersurface when the ambient weighted scalar curvature is bounded from below by a positive constant. In particular, we derive a non-existence result for a weighted stable hypersurface.

Abstract

In the original article [1], Theorem 1.2 (Künneth theorem) is incorrect.

We discuss regularity statements for equidistant decompositions of Riemannian manifolds and for the corresponding quotient spaces. We show that any stratum of the quotient space has curvature locally bounded from both sides.

Abstract

In this work, we extend classical results for subgraphs of functions of bounded variation in (mathbb R^ntimes mathbb R) to the setting of ({textsf{X}}times mathbb R) , where ({textsf{X}}) is an ({textrm{RCD}}(K,N)) metric measure space. In particular, we give the precise expression of the push-forward onto ({textsf{X}}) of the perimeter measure of the subgraph in ({textsf{X}}times mathbb R) of a ({textrm{BV}}) function on ({textsf{X}}) . Moreover, in properly chosen good coordinates, we write the precise expression of the normal to the boundary of the subgraph of a ({textrm{BV}}) function f with respect to the polar vector of f, and we prove change-of-variable formulas.

We study critical points of natural functionals on various spaces of almost Hermitian structures on a compact manifold (M^{2n}). We present a general framework, introducing the notion of gradient of an almost Hermitian functional. As a consequence of the diffeomorphism invariance, we show that a Schur’s type theorem still holds for general almost Hermitian functionals, generalizing a known fact for Riemannian functionals. We present two concrete examples, the Gauduchon’s functional and a close relative of it. These functionals have been studied previously, but not in the most general setup as we do here, and we make some new observations about their critical points.

Let ((M,H,g_H;g)) be a sub-Riemannian manifold and (N, h) be a Riemannian manifold. For a smooth map (u: M rightarrow N), we consider the energy functional (E_G(u) = frac{1}{2} int _M[|textrm{d}u_text {H}|^2 - 2,G(u)] textrm{d}V_M), where (textrm{d}u_text {H}) is the horizontal differential of u, (G:Nrightarrow mathbb {R}) is a smooth function on N. The critical maps of (E_G(u)) are referred to as subelliptic harmonic maps with potential G. In this paper, we investigate the existence problem for subelliptic harmonic maps with potentials by a subelliptic heat flow. Assuming that the target Riemannian manifold has nonpositive sectional curvature and the potential G satisfies various suitable conditions, we prove some Eells–Sampson-type existence results when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation.

Abstract

In this paper, we deal with a strongly pseudoconvex almost CR manifold with a CR contraction. We will prove that the stable manifold of the CR contraction is CR equivalent to the Heisenberg group model.