Annals of Global Analysis and Geometry最新文献

We consider the index of a certain non-compact free-boundary minimal surface with boundary on the rotationally symmetric minimal sphere in the Schwarzschild-AdS geometry with (m>0). As in the Schwarzschild case, we show that in dimensions (nge 4), the surface is stable, whereas in dimension three, the stability depends on the value of the mass (m>0) and the cosmological constant (Lambda <0) via the parameter (mu :=msqrt{-Lambda /3}). We show that while for (mu ge tfrac{5}{27}) the surface is stable, there exist positive numbers (mu _0) and (mu _1), with (mu _1<tfrac{5}{27}), such that for (0<mu <mu _0), the surface is unstable, while for all (mu ge mu _1), the index is at most one.

We study the n-bubble problem on (mathbb {R}^1) with a prescribed density function f that is even, radially increasing, and satisfies a log-concavity requirement. Under these conditions, we find that isoperimetric solutions can be identified for an arbitrary number of regions, and that these solutions have a well-understood and regular structure. This generalizes recent work done on the density function (|x |^p) and stands in contrast to log-convex density functions which are known to have no such regular structure.

In this paper, we formulate and prove a general compactness theorem for harmonic maps of Riemann surfaces using Deligne–Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex curves, there is a family of curves and a subsequence such that both the domains and the maps converge with the singular set consisting of only “non-regular” nodes. This provides a sufficient condition for a neck having zero energy and zero length. As a corollary, the following known fact can be proved: If all domains are diffeomorphic to (S^2), both energy identity and zero distance bubbling hold.

In this note, we study the Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in ({text {CAT}}(1)) space. Under the setting, we prove that the Korevaar–Schoen energy admits a unique minimizer.

We present a geometric construction and characterization of 2n-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson–Walker metric construction for n-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive a complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman–Graham ambient metric and show the vanishing of the Q-curvature.

We show that if an Alexandrov space X has an Alexandrov subspace ({bar{Omega }}) of the same dimension disjoint from the boundary of X, then the topological boundary of ({bar{Omega }}) coincides with its Alexandrov boundary. Similarly, if a noncollapsed ({{,textrm{RCD},}}(K,N)) space X has a noncollapsed ({{,textrm{RCD},}}(K,N)) subspace ({bar{Omega }}) disjoint from boundary of X and with mild boundary condition, then the topological boundary of ({bar{Omega }}) coincides with its De Philippis–Gigli boundary. We then discuss some consequences about convexity of such type of equivalence.

The Hadamard expansion describes the singularity structure of Green’s operators associated with a normally hyperbolic operator P in terms of Riesz distributions (fundamental solutions on Minkowski space, transported to the manifold via the exponential map) and Hadamard coefficients (smooth sections in two variables, corresponding to the heat Kernel coefficients in the Riemannian case). In this paper, we derive an asymptotic expansion analogous to the Hadamard expansion for powers of advanced/retarded Green’s operators associated with P, as well as expansions for advanced/retarded Green’s operators associated with (P-z) for (zin mathbb {C}). These expansions involve the same Hadamard coefficients as the original Hadamard expansion, as well as the same or analogous (with built-in z-dependence) Riesz distributions.

We construct explicit complex-valued p-harmonic functions and harmonic morphisms on the classical compact symmetric complex and quaternionic Grassmannians. The ingredients for our construction method are joint eigenfunctions of the classical Laplace–Beltrami and the so-called conformality operator. A known duality principle implies that these p-harmonic functions and harmonic morphisms also induce such solutions on the Riemannian symmetric non-compact dual spaces.

We prove a general estimate for the Weyl remainder of an elliptic, semiclassical pseudodifferential operator in terms of volumes of recurrence sets for the Hamilton flow of its principal symbol. This quantifies earlier results of Volovoy (Comm Partial Differential Equations 15:1509–1563, 1990; Ann Global Anal Geom 8:127–136, 1990). Our result particularly improves Weyl remainder exponents for compact Lie groups and surfaces of revolution. And gives a quantitative estimate for Bérard’s Weyl remainder in terms of the maximal expansion rate and topological entropy of the geodesic flow.

The object of this article is to study a new class of almost contact metric structures which are integrable but non normal. Secondly, we explain a method of construction for normal manifold starting from a non-normal but integrable manifold. Illustrative examples are given.