Annals of Global Analysis and Geometry最新文献

Let (M, L) be a (compact) non-spin spin(^c) manifold. Fix a Riemannian metric g on M and a connection A on L, and let (D_L) be the associated spin(^c) Dirac operator. Let (R^{text {tw }}_{(g,A)}:=R_g + 2ic(Omega )) be the twisted scalar curvature (which takes values in the endomorphisms of the spinor bundle), where (R_g) is the scalar curvature of g and (2ic(Omega )) comes from the curvature 2-form (Omega ) of the connection A. Then the Lichnerowicz-Schrödinger formula for the square of the Dirac operator takes the form (D_L^2 =nabla ^*nabla + frac{1}{4}R^{text {tw }}_{(g,A)}). In a previous work we proved that a closed non-spin simply-connected spin(^c)-manifold (M, L) of dimension (nge 5) admits a pair (g, A) such that (R^{text {tw }}_{(g,A)}>0) if and only if the index (alpha ^c(M,L):={text {ind}}D_L) vanishes in (K_n). In this paper we introduce a scalar-valued generalized scalar curvature (R^{text {gen }}_{(g,A)}:=R_g - 2|Omega |_{op}), where (|Omega |_{op}) is the pointwise operator norm of Clifford multiplication (c(Omega )), acting on spinors. We show that the positivity condition on the operator (R^{text {tw }}_{(g,A)}) is equivalent to the positivity of the scalar function (R^{text {gen }}_{(g,A)}). We prove a corresponding trichotomy theorem concerning the curvature (R^{text {gen }}_{(g,A)}), and study its implications. We also show that the space (mathcal {R}^{{textrm{gen}+}}(M,L)) of pairs (g, A) with (R^{text {gen }}_{(g,A)}>0) has non-trivial topology, and address a conjecture about non-triviality of the “index difference” map.

On a compact Riemannian manifold M with boundary Y, we express the log of the zeta-determinant of the Dirichlet-to-Neumann operator acting on q-forms on Y as the difference of the log of the zeta-determinant of the Laplacian on q-forms on M with the absolute boundary condition and that of the Laplacian with the Dirichlet boundary condition with an additional term which is expressed by curvature tensors. When the dimension of M is 2 and 3, we compute these terms explicitly. We also discuss the value of the zeta function at zero associated to the Dirichlet-to-Neumann operator by using a metric rescaling method. As an application, we recover the result of the conformal invariance obtained in Guillarmou and Guillope (Int Math Res Not IMRN 2007(22):rnm099, 2007) when ({text {dim}}M = 2).

We consider the problem of finding conformal metrics on the standard half sphere with prescribed scalar curvature and zero-boundary mean curvature. We prove a perturbation result when the curvature function is flat near its boundary critical points. As a product we extend some previous well known results and provide an entirely new one.

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).