Domains In Manifolds Research Articles

Generalized Reilly's identity and its applications in V-static manifolds

In this paper, we explore the applications of a Reilly-type identity within the context of V-static manifolds. These manifolds are characterized by being Riemannian manifolds with a boundary, possessing constant scalar curvature, and featuring a metric that serves as a critical point for the volume functional (with a fixed metric on the boundary). We apply a Reilly-type identity due Li-Xia within this specific framework and leverage it to establish Heintze-Karcher and Minkowski inequalities for bounded domains in such manifolds. Additionally, we delve into the rigidity of both results in scenarios where equality holds.

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

Kobayashi complete domains in complex manifolds

In this paper, we give sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains in complex manifolds. The first result gives a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. This follows from our second result, which may be of independent interest, in a much more general setting. This extends a result of Gaussier to the setting of manifolds.

Learning a target-dependent classifier for cross-domain semantic segmentation: Fine-tuning versus meta-learning

Recently proposed domain adaptation arts have dominated the field of cross-domain semantic segmentation by operating domain manifolds alignment and learning an optimal joint hypothesis (joint-domain classifier) for both source and target domains. However, a joint-domain classifier can still violate the cluster assumption in the target domain in case domain manifolds are not fully aligned after domain adaptation. In this work, we raise the intractability of perfect domain alignment and turn to exploit a novel hypothesis: a target-dependent classifier, to efficiently adapt to the target domain clusters even given a certain degree of domain misalignment. Specifically, we first propose an unsupervised fine-tuning strategy, which optimizes the joint hypothesis of vanilla domain adaptation into a target-dependent hypothesis to better fit with the target domain clusters. Second, we connect the “learning to learn” concept of meta-learning with pixel-wise domain adaptation, which serves as a reliable hypothesis initialization, providing an alternative solution to learning a more generalized target-dependent classifier. The proposed learning method is general to conventional domain adaptation models. In experiments, we recycle the pre-trained conventional DA models and learn target-dependent classifiers with the proposed method. Experimental results on synthetic-to-real adaptation and cross-city adaptation benchmarks demonstrate that the target-dependent classifier leads over state-of-the-art performance.

Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian manifolds

Abstract We show that there exists a nonconstant CR-holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative pseudohermitian bisectional curvature with the CR maximal volume growth property. This is the very first step toward the CR analogue of the Yau uniformization conjecture which states that any complete noncompact Sasakian manifold of positive pseudohermitian bisectional curvature is CR biholomorphic to the standard Heisenberg group. More precisely, we first construct CR-holomorphic functions with controlled growth in a sequence of exhaustion domains in Sasakian manifolds by applying the Cheeger–Colding theory. Secondly, via the CR analogue of a tangent cone at infinity and the three-circle theorem, we are able to take the subsequence to obtain a nonconstant CR-holomorphic function of polynomial growth.

A class of weighted isoperimetric inequalities in hyperbolic space

In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia [Trans. Amer. Math. Soc. 372 (2019), pp. 6771–6803]. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzschild manifold.

Around L1 (un)boundedness of Bergman and Szegö projections

We consider the problem of L1 (un)boundedness for a wide class of orthogonal projections, including Bergman projections on domains in complex manifolds and Szegö projections on abstract CR manifolds.

Domains of discontinuity in oriented flag manifolds

We study actions of discrete subgroups Γ $\Gamma$ of semi-simple Lie groups G $G$ on associated oriented flag manifolds. These are quotients G / P $G/P$ , where the subgroup P $P$ lies between a parabolic subgroup and its identity component. For Anosov subgroups Γ ⊂ G $\Gamma \subset G$ , we identify domains in oriented flag manifolds by removing a set obtained from the limit set of Γ $\Gamma$ , and give a combinatorial description of proper discontinuity and cocompactness of these domains. This generalizes analogous results of Kapovich–Leeb–Porti to the oriented setting. We give first examples of cocompact domains of discontinuity which are not lifts of domains in unoriented flag manifolds. These include in particular domains in oriented Grassmannians for Hitchin representations, which we also show to be nonempty. As a further application of the oriented setup, we give a new lower bound on the number of connected components of B $B$ -Anosov representations of a closed surface group into SL ( n , R ) $\operatorname{SL}(n,\mathbb {R})$ .

Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds

In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock (An estimate on the Hessian of the heat kernel: 355–371, 1996) and removes in particular the compact manifold restriction. To demonstrate the potential of these formulas, we give as application explicit quantitative local estimates for the Hessian of the heat semigroup, as well as for harmonic functions on regular domains in Riemannian manifolds.

Topological Properties of Regular Boundary Domains in Riemannian Manifolds

Topological Properties of Regular Boundary Domains in Riemannian Manifolds

Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space [Formula: see text], we have The first (positive) Dirichlet eigenfunction of the Laplacian on a ball in [Formula: see text] raised to some power [Formula: see text] is strictly concave; Let [Formula: see text] be the heat kernel on [Formula: see text]. Then [Formula: see text] is strictly log-concave in [Formula: see text] for [Formula: see text] and [Formula: see text].

Euclidean domains in complex manifolds

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets K∪M in a complex manifold X, where K is a compact holomorphically convex set in an open Stein neighbourhood, M is an embedded Stein submanifold, and K∩M is compact O(M)-convex. We prove a Docquier–Grauert type theorem concerning biholomorphic equivalence of neighbourhoods of such sets, and we give sufficient conditions for the existence of Stein neighbourhoods of K∪M, biholomorphic to domains in Cn with n=dim⁡X, such that M is mapped onto a closed complex submanifold of Cn.

On the Negativity of Ricci Curvatures of Complete Conformal Metrics

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases.

Holomorphic approximation via Dolbeault cohomology

The purpose of this paper is to study holomorphic approximation and approximation of $$\overline{\partial }$$ -closed forms in complex manifolds of complex dimension $$n\ge 1$$ . We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the $${{\mathcal {C}}}^\infty $$ -smooth and the $$L^2$$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.

An integral formula and its applications on sub-static manifolds

In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \cite{Re2} and the recent result from \cite{QX}. It provides a robust tool for sub-static manifolds regardless of the underlying topology. Using this formula and suitable elliptic PDEs, we prove Heintze-Karcher type inequalities for bounded domains in general sub-static manifolds which recovers some of the results from Brendle \cite{Br} as special cases. On the other hand, we prove a Minkowski inequality for static convex hypersurfaces in a sub-static warped product manifold. Moreover, we obtain an almost Schur lemma for horo-convex hypersurfaces in the hyperbolic space and convex hypersurfaces in the hemi-sphere, which can be viewed as a special Alexandrov-Fenchel inequality.

The local invariant for scale structures on mapping spaces

A scale Hilbert space is a natural generalization of a Hilbert space which considers not only a single Hilbert space but a nested sequence of subspaces. Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder as a new concept of smooth structures in infinite dimensions. In this paper, we prove that scale structures on mapping spaces are completely determined by the dimension of domain manifolds. We also give a complete description of the local invariant introduced by U. Frauenfelder for these spaces. Product mapping spaces and relative mapping spaces are also studied. Our approach is based on the spectral resolution of Laplace type operators together with the eigenvalue growth estimate.

Finitely Additive Measures on the Unstable Leaves of Anosov Diffeomorphisms

We obtain a qualitative characterization of the convergence rate of the averages (with respect to the Margulis measure) of C2 functions over the iterations of domains in unstable manifolds of a topologically mixing C3 Anosov diffeomorphism with oriented invariant foliations. For this purpose, we extend the constructions of Margulis and Bufetov and introduce holonomy invariant families of finitely additive measures on unstable leaves and a Banach space in which holonomy invariant measures correspond to the (generalized) eigenfunctions of the transfer operator with biggest eigenvalues.

Rigidity of convex divisible domains in flag manifolds

In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of $p$-planes in $\mathbb{R}^{2p}$ when $p > 1$. Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group SO$(p, p)$.

Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds

The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with $$L^{\infty }$$ -variable coefficients in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in $$L^2$$ -based Sobolev spaces by using the remarkable Necas–Babuska–Brezzi technique (see Babuska in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Necas in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.

An upper bound for the second Neumann eigenvalue on Riemannian manifolds

The Szego–Weinberger inequality asserts that the second Neumann eigenvalue $$\mu _2$$ of Laplace operator on a bounded domain in $$\mathbb {R}^n$$ is bounded from above by that of a ball of the same volume. In this note, we prove an upper bound for $$\mu _2$$ on domains in Riemannian manifolds, which can be viewed as a Sezgo–Weinberger type inequality.