Annals of Global Analysis and Geometry最新文献

In this work, we will study the boundary behaviors of a spacelike positive constant mean curvature surface (Sigma ) in the Schwarzschild spacetime exterior to the black hole. We consider two boundaries: the future null infinity (mathcal {I}^+) and the horizon. Suppose near (mathcal {I}^+), (Sigma ) is the graph of a function (-P(textbf{y},s)) in the form (overline{v}=-P), where (overline{v}) is the retarded null coordinate with (s=r^{-1}) and (textbf{y}in mathbb {S}^2). Suppose the boundary value of (P(textbf{y},s)) at (s=0) is a smooth function f on the unit sphere (mathbb {S}^2). If P is (C^4) at (mathcal {I}^+), then f must satisfy a fourth order PDE on (mathbb {S}^2). If P is (C^3), then all the derivatives of P up to order three can be expressed in terms of f and its derivatives on (mathbb {S}^2). For the extrinsic geometry of (Sigma ), under certain conditions we obtain decay rate of the trace-free part of the second fundamental forms (mathring{A}). In case (mathring{A}) decays fast enough, some further restrictions on f are given. For the intrinsic geometry, we show that under certain conditions, (Sigma ) is asymptotically hyperbolic in the sense of Chruściel–Herzlich (Pac J Math 212(2):231–264, 2003). Near the horizon, we prove that under certain conditions, (Sigma ) can be expressed as the graph of a function u which is smooth in (eta =left( 1-frac{2m}{r}right) ^{frac{1}{2}}) and (textbf{y}in mathbb {S}^2), and all its derivatives are determined by the boundary value u at (eta =0). In particular, a Neumann-type condition is obtained. This may be related to a remark of Bartnik (in: Proc Centre Math Anal Austral Nat Univ, 1987). As for intrinsic geometry, we show that under certain conditions the inner boundary of (Sigma ) given by (eta =0) is totally geodesic.

We prove some vanishing theorems for the Kohn–Rossi cohomology of some spherical CR manifolds. To this end, we use a canonical contact form defined via the Patterson–Sullivan measure and Weitzenböck-type formulae for the Kohn Laplacian. We also see that our results are optimal in some cases.

We discuss in detail Alan Schoen’s I-WP surface, an embedded triply periodic minimal surface of genus 4 with cubical symmetries. We exhibit various geometric realizations of this surface with the same conformal structure and use them to prove that the associate family of the I-WP surface contains six surfaces congruent to I-WP at Bonnet angles that are multiples of (60^circ ).

We show that under some natural geometric assumption, Einstein metrics on conformal products of two compact conformal manifolds are warped product metrics.

In this paper, we study conformal solitons for the mean curvature flow in hyperbolic space (mathbb {H}^{n+1}). Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field (-partial _0). We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability and address the case of non-compact boundaries contained between two parallel hyperplanes of (partial _infty mathbb {H}^{n+1}). We conclude by proving rigidity results for bowl and grim-reaper cylinders.

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.