Annals of Global Analysis and Geometry最新文献

We study the homology groups of certain 2-connected 7-manifolds admitting quasi-regular Sasaki–Einstein metrics, among them, we found 52 new examples of Sasaki–Einstein rational homology 7-spheres, extending the list given by Boyer et al. (Ann Inst Fourier 52(5):1569–1584, 2002). As a consequence, we exhibit new families of positive Sasakian homotopy 9-spheres given as cyclic branched covers, determine their diffeomorphism types and find out which elements do not admit extremal Sasaki metrics. We also improve previous results given by Boyer (Note Mat 28:63–105, 2008) showing new examples of Sasaki–Einstein 2-connected 7-manifolds homeomorphic to connected sums of (S^3times S^4). Actually, we show that manifolds of the form (#kleft( S^{3} times S^{4}right) ) admit Sasaki–Einstein metrics for 22 different values of k. All these links arise as Thom–Sebastiani sums of chain type singularities and cycle type singularities where Orlik’s conjecture holds due to a recent result by Hertling and Mase (J Algebra Number Theory 16(4):955–1024, 2022).

We apply the Berglund–Hübsch transpose rule from BHK mirror symmetry to show that to an (n-1)-dimensional Calabi–Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension (2n+1) which are (n-1)-connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exist four seven-dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki–Einstein.

Considering a Monge-Ampère equation with prescribed affine Gauss curvature, we first show the completeness of centroaffine metric on the convex domain and derive a gradient estimate of the convex solution and then give different orders of two eigenvalues of the Hessian with respect to the distance function. We also show that the curvature of level sets of the convex solution is uniformly bounded, and show that there exist a class of Euclidean-complete hyperbolic surfaces with prescribed affine Gauss curvature and with bounded affine principal curvatures.

Pseudo-Riemannian manifolds with nonzero parallel Weyl tensor which are not locally symmetric are known as ECS manifolds. Every ECS manifold carries a distinguished null parallel distribution (mathcal {D}), the rank (din {1,2}) of which is referred to as the rank of the manifold itself. Under a natural genericity assumption on the Weyl tensor, we fully describe the universal coverings of compact rank-one ECS manifolds. We then show that any generic compact rank-one ECS manifold must be translational, in the sense that the holonomy group of the natural flat connection induced on (mathcal {D}) is either trivial or isomorphic to ({mathbb {Z}}_2). We also prove that all four-dimensional rank-one ECS manifolds are noncompact, this time without having to assume genericity, as it is always the case in dimension four.

We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (Differ Geom Appl 19:193–210, 2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of analytic properties for this flow, such as uniqueness, smoothness, short-time existence, and some sufficient conditions for long-time existence. This description potentially subsumes a large class of geometric PDE problems from different contexts. As applications, we recover and unify a number of results in the literature: for the isometric flow of (text {G}_2)-structures, by Grigorian (Adv Math 308:142–207, 2017; Calculas Variat Partial Differ Equ 58:157, 2019), Bagaglini (J Geom Anal, 2009), and Dwivedi-Gianniotis-Karigiannis (J Geom Anal 31(2):1855-1933, 2021); and for harmonic almost complex structures, by He (Energy minimizing harmonic almost complex structures, 2019) and He-Li (Trans Am Math Soc 374(9):6179–6199, 2021). Our theory also establishes original properties regarding harmonic flows of parallelisms and almost contact structures.

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.