Abstract: In the author’s previous joint work with Hein (Commun Math Phys 347:183–221, 2016), a mass formula for asymptotically locally Euclidean Kähler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension 4 presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chruściel fall-off conditions that sufficed in higher dimensions. Nevertheless, the present article shows that techniques of four-dimensional symplectic geometry can be used to obtain all the major results of Hein-LeBrun (2016), assuming only Chruściel-type fall-off. In particular, the present article presents a new proof of our Penrose-type inequality for the mass of an asymptotically Euclidean Kähler manifold that only requires this very weak metric fall-off. PubDate: 2019-03-21
Abstract: In this paper, we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three-dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satisfies a positive lower bound depending only on the ambient curvatures. We conclude the paper proving that a stable, compact without boundary, nonzero constant mean curvature surface in a general Riemannian is a topological sphere provided its mean curvature has a lower bound depending only on the scalar curvature of the ambient space and the squared norm of the mean curvature vector field of the immersion of the ambient space in some Euclidean space. PubDate: 2019-03-07
Abstract: In this paper we prove existence and uniqueness of the Björling problem for the class of immersed surfaces in \(\mathbb {R}^3\) whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Möbius strip for an arbitrary large class of prescribed functions. In particular, we use the Björling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Möbius strip in \(\mathbb {R}^3\) . PubDate: 2019-03-02
Abstract: We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces \({\mathbb {L}}(\kappa ,\tau )\) with isometry group of dimension 4, which are dual to the Abresch–Rosenberg differentials in the Riemannian counterparts \({\mathbb {E}}(\kappa ,\tau )\) , and obtain some consequences. On the one hand, we classify explicitly those surfaces in \({\mathbb {L}}(\kappa ,\tau )\) with zero differential. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature. PubDate: 2019-03-01
Abstract: Let M be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on M to a small tubular neighborhood of each connected component of its singular stratum is foliated diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes characteristic numbers of M as the sum of residues associated with the infinitesimal foliation at the components of the singular stratum of the flow. PubDate: 2019-03-01
Abstract: We give a short proof of a strong version of the short-time asymptotic expansion of heat kernels associated with Laplace-type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this to show that repeated convolution of the approximate heat kernels can be used to approximate the heat kernel on all of M, which is related to expressing the heat kernel as a path integral. This scheme is then applied to obtain a short-time asymptotic expansion of the heat kernel at the cut locus. PubDate: 2019-03-01
Abstract: We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia–Szegö-, Alexandrov–Fenchel- and Penrose-type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary. PubDate: 2019-03-01
Abstract: Given a compact Riemannian manifold (M, g) and two positive functions \(\rho \) and \(\sigma \) , we are interested in the eigenvalues of the Dirichlet energy functional weighted by \(\sigma \) , with respect to the \(L^2\) inner product weighted by \(\rho \) . Under some regularity conditions on \(\rho \) and \(\sigma \) , these eigenvalues are those of the operator \( -\rho ^{-1} \text{ div }(\sigma \nabla u) \) with Neumann conditions on the boundary if \(\partial M\ne \emptyset \) . We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved. PubDate: 2019-03-01
Abstract: The aim of this article is to study expansions of solutions to an extremal metric type equation on the blowup of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blowups of extremal Kähler surfaces (Székelyhidi in Duke Math J 161(8):1411–1453, 2012). PubDate: 2019-03-01
Abstract: We study a cohomology theory \(H^{\bullet }_{\varphi }\) , which we call the \({\mathcal {L}}_B\) -cohomology, on compact torsion-free \(\mathrm {G}_2\) manifolds. We show that \(H^k_{\varphi } \cong H^k_{\mathrm {dR}}\) for \(k \ne 3, 4\) , but that \(H^k_{\varphi }\) is infinite-dimensional for \(k = 3,4\) . Nevertheless, there is a canonical injection \(H^k_{\mathrm {dR}} \rightarrow H^k_{\varphi }\) . The \({\mathcal {L}}_B\) -cohomology also satisfies a Poincaré duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative \(\mathrm {d}\) and the derivation \({\mathcal {L}}_B\) and uses both Hodge theory and the special properties of \(\mathrm {G}_2\) -structures in an essential way. As an application of our results, we prove that compact torsion-free \(\mathrm {G}_2\) manifolds are ‘almost formal’ in the sense that most of the Massey triple products necessarily must vanish. PubDate: 2019-03-01
Abstract: We introduce in this paper an adjustment for the Webster–Tanaka connection of pseudo-Hermitian geometry on CR manifolds, which behaves naturally with respect to issues concerning parallel sections and holonomy. We call this connection and its holonomy group basic. The basic holonomy group is suitable to describe pseudo-Einstein spaces. Moreover, for pseudo-Hermitian spin manifolds, we discuss the existence of (transversally) parallel spinors. Such spinors exist on pseudo-Einstein spaces, no matter of the sign of the Webster scalar curvature. PubDate: 2019-03-01
Abstract: Given a compact Lie subgroup G of the isometry group of a compact Riemannian manifold M with a Riemannian connection \(\nabla ,\) a G-symmetrization process of a vector field of M is introduced and it is proved that the critical points of the energy functional $$\begin{aligned} F(X):=\frac{\int _{M}\left\ \nabla X\right\ ^{2}\mathrm{d}M}{\int _{M}\left\ X\right\ ^{2}\mathrm{d}M} \end{aligned}$$ on the space of \(\ G\) -invariant vector fields are critical points of F on the space of all vector fields of M and that this inclusion may be strict in general. One proves that the infimum of F on \({\mathbb {S}}^{3}\) is not assumed by a \({\mathbb {S}}^{3}\) -invariant vector field. It is proved that the infimum of F on a sphere \({\mathbb {S}}^{n},\) \(n\ge 2,\) of radius 1 / k, is \(k^{2},\) and is assumed by a vector field invariant by the isotropy subgroup of the isometry group of \({\mathbb {S}}^{n}\) at any given point of \({\mathbb {S}} ^{n}\) . It is proved that if G is a compact Lie subgroup of the isometry group of a compact rank 1 symmetric space M which leaves pointwise fixed a totally geodesic submanifold of dimension bigger than or equal to 1, then the infimum of F is assumed by a G-invariant vector field. PubDate: 2019-03-01
Abstract: In this paper, we first investigate the flow of convex surfaces in the space form \(\mathbb {R}^3(\kappa )~(\kappa =0,1,-1)\) expanding by \(F^{-\alpha }\) , where F is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power \(\alpha \in (0,1]\) for \(\kappa =0,-1\) and \(\alpha =1\) for \(\kappa =1\) . By deriving that the pinching ratio of the flow surface \(M_t\) is no greater than that of the initial surface \(M_0\) , we prove the long time existence and the convergence of the flow. No concavity assumption of F is required. We also show that for the flow in \(\mathbb {H}^3\) with \(\alpha \in (0,1)\) , the limit shape may not be necessarily round after rescaling. PubDate: 2019-03-01
Abstract: In the present paper, we first consider the weighted eigenvalue problem \(\Delta _f u=\lambda _{f}u\) in M with the Robin boundary condition \(\frac{\partial u}{\partial \nu }+\beta u=0\) on \(\partial M\) , where \((M^n,g,e^{-f})\) is a compact n-dimensional weighted Riemannian manifold of nonnegative Bakry–Émery Ricci curvature. We derive under some convexity condition of the boundary \(\partial M\) , an explicit lower bound of the first weighted Robin eigenvalue \(\lambda _{1,f}(\beta )\) depending only on the geometry of M and the constant \(\beta \) appearing in the boundary condition. Another new estimate for \(\lambda _{1,f}(\beta )\) with respect to the first nonzero Neumann eigenvalue \(\mu _{2,f}\) of the weighted Laplacian \(\Delta _f\) is also obtained. Furthermore, we provide some lower bounds for the first buckling and clamped plate eigenvalues of the bi-drifting Laplacian on weighted manifolds. PubDate: 2019-02-23
Abstract: We investigate isometric immersions of locally conformally Kähler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces. PubDate: 2019-02-21
Abstract: We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6, we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinite singularities, we can provide these wrinkled fibrations in dimension 6 with near-symplectic structures. PubDate: 2019-02-13
Abstract: In this article, we consider \(L^{2}\) harmonic forms on a complete non-compact Riemannian manifold X with a nonzero parallel form \(\omega \) . The main result is that if \((X,\omega )\) is a complete \(G_{2}\) - (or \(\textit{Spin}(7)\) -) manifold with a d(linear) \(G_{2}\) - (or \(\textit{Spin}(7)\) -) structure form \(\omega \) , then the \(L^{2}\) harmonic 2-forms on X vanish. As an application, we prove that the instanton equation with square-integrable curvature on \((X,\omega )\) only has trivial solution. We would also consider the Hodge theory on the principal G-bundle E over \((X,\omega )\) . PubDate: 2019-02-13
Abstract: The Besse’s conjecture was posed on the well-known book Einstein manifolds by Arthur L. Besse, which describes critical points of Hilbert–Einstein functional with constraint of unit volume and constant scalar curvature. In this article, we show that there is an interesting connection between Besse’s conjecture and positive mass theorem for Brown–York mass. With the aid of positive mass theorem, we investigate the geometric structure of CPE manifolds and this provides us further understandings about Besse’s conjecture. As a related topic, we also discuss corresponding results for V-static metrics. PubDate: 2019-02-11
Abstract: We study the low-regularity (in-)extendibility of spacetimes within the synthetic-geometric framework of Lorentzian length spaces developed in Kunzinger and Sämann (Ann Glob Anal Geom 54(3):399–447, 2018). To this end, we introduce appropriate notions of geodesics and timelike geodesic completeness and prove a general inextendibility result. Our results shed new light on recent analytic work in this direction and, for the first time, relate low-regularity inextendibility to (synthetic) curvature blow-up. PubDate: 2019-02-01
Abstract: We introduce a notion of equivalence for singular foliations—understood as suitable families of vector fields—that preserves their transverse geometry. Associated with every singular foliation, there is a holonomy groupoid, by the work of Androulidakis–Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence, we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980s. PubDate: 2019-02-01
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