Annals of Global Analysis and Geometry最新文献

In this work we investigate Gromov–Hausdorff limits of compact surfaces carrying length metrics. More precisely, we consider the case where all surfaces have the same Euler characteristic. We give a complete description of the limit spaces and study their topological properties. Our investigation builds on the results of a previous work which treats the case of closed surfaces.

In this paper we focus on the interplay between the behaviour of the Frölicher spectral sequence and the existence of special Hermitian metrics on the manifold, such as balanced, SKT or generalized Gauduchon. The study of balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex structures allows us to provide infinite families of compact balanced manifolds with Frölicher spectral sequence not degenerating at the second page. Moreover, this result is extended to non-degeneration at any arbitrary page. Similar results are obtained for the Frölicher spectral sequence of compact generalized Gauduchon manifolds. We also find a compact SKT manifold whose Frölicher spectral sequence does not degenerate at the second page, thus providing a counterexample to a conjecture by Popovici.

In this work we construct new multi-dimensional families of compact minimal submanifolds of the classical Riemannian symmetric spaces ({{textbf {S}}U}(n)/textbf{SO}(n)), ({{textbf {S}}p}(n)/{{textbf {U}}}(n)), (textbf{SO}(2n)/{{textbf {U}}}(n)) and ({{textbf {S}}U}(2n)/{{textbf {S}}p}(n)) of codimension two.

In this paper, we prove parabolic Schauder estimates for the Laplace-Beltrami operator on a manifold M with fibered boundary and a (Phi )-metric (g_Phi ). This setting generalizes the asymptotically conical (scattering) spaces and includes special cases of gravitational instantons. This paper, combined with part II, lay the crucial groundwork for forthcoming discussions on geometric flows in this setting; especially the Yamabe- and mean curvature flow.

Let X denote a noncompact finite volume hyperbolic Riemann surface of genus (gge 2), with only one puncture at (iinfty ) (identifying X with its universal cover ({mathbb {H}})). Let ({{{overline{X}}}}:=Xcup lbrace iinfty rbrace ) denote the Satake compactification of X. Let (Omega _{{{{overline{X}}}}}) denote the cotangent bundle on ({{{overline{X}}}}). For (kgg 1), we derive an estimate for (mu _{{ {overline{X}}}}^{textrm{Ber},{{k}}}), the Bergman metric associated to the line bundle ({{mathcal {L}}}^{k}:=Omega _{{{{overline{X}}}}}^{otimes {{k}}}otimes {{mathcal {O}}}_{{{{overline{X}}}}}((k-1)iinfty )). For a given (dge 1), the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by (mu _{textrm{Sym}^{{d}}({{overline{X}}})}^{textrm{FS},k}), defines a Kähler metric on (textrm{Sym}^{{d}}({{overline{X}}})), the d-fold symmetric product of ({{{overline{X}}}}). Using our estimates of (mu _{{ {overline{X}}}}^{textrm{Ber},{{k}}}), as an application, we derive an estimate for (mu _{textrm{Sym}^{{d}}({{overline{X}}}),textrm{vol}}^{textrm{FS},k}), the volume form associated to the (1,1)-form (mu _{textrm{Sym}^{{d}}({{overline{X}}})}^{textrm{FS},k}).

We study the effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes, both in the spacelike and the timelike case, respectively. In the spacelike case, we study totally umbilic warped product graphs and give a full characterization of embedded surfaces with constant spacetime mean curvature using an Alexandrov Theorem by Brendle and Borghini–Fogagnolo–Pinamonti. In the timelike case, we achieve a characterization of photon surfaces with constant umbilicity factor similar to a result by Cederbaum–Galloway.

In this paper, an Alexandrov–Fenchel inequality is established for closed 2-convex spacelike hypersurface in de Sitter space by investigating the behavior of some locally constrained inverse curvature flow, which provides a partial answer to the conjecture raised by Hu and Li in (Adv Math 413:108826, 2023).

In this paper we establish a comparison formula of the absolute and relative real analytic torsion forms over fibrations with boundaries. The key tool is a gluing formula of analytic torsion forms proved by Puchol and Zhang and the author. As a consequence of the comparison formula, we prove another version of the gluing formula of the analytic torsion forms conjectured by the author.

As an application of the general theory on extrinsic geometry (Doubrov et al. in SIGMA Symmetry Integr Geom Methods Appl 17:061, 2021), we investigate extrinsic geometry in flag varieties and systems of linear PDE’s for a class of special interest associated with the adjoint representation of (mathfrak {sl}(3)). We carry out a complete local classification of the homogeneous structures in this class. As a result, we find 7 kinds of new systems of linear PDE’s of second order on a 3-dimensional contact manifold each of which has a solution space of dimension 8. Among them there are included a system of PDE’s called contact Cayley’s surface and one which has (varvec{mathfrak {sl}}(2)) symmetry.