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.

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.

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.

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.

The radial map u(x) (=) (frac{x}{Vert xVert }) is a well-known example of a harmonic map from ({mathbb {R}}^m,-,{0}) into the spheres ({mathbb {S}}^{m-1}) with a point singularity at x (=) 0. In Nakauchi (Examples Counterexamples 3:100107, 2023), the author constructed recursively a family of harmonic maps (u^{(n)}) into ({mathbb {S}}^{m^n-1}) with a point singularity at the origin ((n = 1,,2,ldots )), such that (u^{(1)}) is the above radial map. It is known that for m (ge ) 3, the radial map (u^{(1)}) is not only stable as a harmonic map but also a minimizer of the energy of harmonic maps. In this paper, we show that for n (ge ) 2, (u^{(n)}) may be unstable as a harmonic map. Indeed we prove that under the assumption n > ({displaystyle frac{sqrt{3}-1}{2},(m-1)}) ((m ge 3), (n ge 2)), the map (u^{(n)}) is unstable as a harmonic map. It is remarkable that they are unstable and our result gives many examples of unstable harmonic maps into the spheres with a point singularity at the origin.

We continue our investigation of the interplay between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory. We adopt the perspective that the geometric implementation of the modular group is given by the flow generated by an Euler element of the Lie algebra (an element defining a 3-grading). Since any Euler element of a semisimple Lie algebra specifies a canonical non-compactly causal symmetric space (M = G/H), we turn in this paper to the geometry of this flow. Our main results concern the positivity region W of the flow (the corresponding wedge region): If G has trivial center, then W is connected, it coincides with the so-called observer domain, specified by a trajectory of the modular flow which at the same time is a causal geodesic. It can also be characterized in terms of a geometric KMS condition, and it has a natural structure of an equivariant fiber bundle over a Riemannian symmetric space that exhibits it as a real form of the crown domain of G/K. Among the tools that we need for these results are two observations of independent interest: a polar decomposition of the positivity domain and a convexity theorem for G-translates of open H-orbits in the minimal flag manifold specified by the 3-grading.