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The Convexity of Entire Spacelike Hypersurfaces with Constant $$\sigma _{n-1}$$ Curvature in Minkowski Space J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-20 Changyu Ren, Zhizhang Wang, Ling Xiao
We prove that, in Minkowski space, if a spacelike, \((n-1)\)-convex hypersurface \(\mathcal {M}\) with constant \(\sigma _{n-1}\) curvature has bounded principal curvatures, then \(\mathcal {M}\) is convex. Moreover, if \(\mathcal {M}\) is not strictly convex, after an \(\mathbb {R}^{n,1}\) rigid motion, \(\mathcal {M}\) splits as a product \(\mathcal {M}^{n-1}\times \mathbb {R}.\)
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The Curvature Operator of the Second Kind in Dimension Three J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-20 Harry Fluck, Xiaolong Li
This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that \(\alpha \)-positive/\(\alpha \)-nonnegative curvature operator of the second kind is preserved
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Remarks on the Relation of Log-Concave and Contoured Distributions in $${\mathbb {R}}^n$$ J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-20 Nikos Dafnis, Antonis Tsolomitis
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Positive Solution for an Elliptic System with Critical Exponent and Logarithmic Terms J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-17 Hichem Hajaiej, Tianhao Liu, Linjie Song, Wenming Zou
In this paper, we study the existence and nonexistence of positive solutions for the following coupled elliptic system with critical exponent and logarithmic terms: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda _{1}u+ \mu _1|u|^{2}u+\beta |v|^{2}u+\theta _1 u\log u^2, &{} \quad x\in \Omega ,\\ -\Delta v=\lambda _{2}v+ \mu _2|v|^{2}v+\beta |u|^{2}v+\theta _2 v\log v^2, &{}\quad x\in
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Uniform Stationary Phase Estimate with Limited Smoothness J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-12 Sanghyuk Lee, Sewook Oh
In this paper, we consider uniform estimates for the oscillatory integrals with stationary phase, which were previously studied by Farah–Rousset–Tzvetkov and Alazard–Burq–Zuily. We significantly reduce the order of required regularity on the phase and amplitude functions for the uniform estimate. We also study estimates for the oscillatory integrals of which phase and amplitude functions depend on
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On the Isometry Group of Immortal Homogeneous Ricci Flows J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-10 Roberto Araujo
We establish a structure result for the isometry group of non-compact, homogeneous manifolds admitting an immortal homogeneous Ricci flow solution.
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The Borel Map for Compact Subanalytic Subsets of $$\mathbb {C}^m$$ J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-10 Paulo D. Cordaro, Giuseppe Della Sala, Bernhard Lamel
We define, for a compact subset K of complex Euclidean space containing the origin, the so-called Borel map (at the origin). We discuss its properties in detail and state, in the case when K is subanalytic, two conjectures relating the injectivity and surjectivity of the Borel map with properties of the polynomial hull of K. We give strong evidence for the validity of the conjectures (e.g. the open
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$$L^{p}$$ Estimates for the Bergman Projection on Generalized Fock Spaces J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-10 Thuc Trong Phung
We obtain \(L^{p}\) bounds for the Bergman projection \(P_{\varphi }\) on \(\mathbb {C}^{n}\) for a class of weights \(\varphi \) whose complex Hessian has comparable eigenvalues. This relies on an extension of the estimate on the Bergman kernel obtained previously by Dall’Ara.
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On Singularities of the Gauss Map Components of Surfaces in $${{\mathbb {R}}}^4$$ J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-09 Wojciech Domitrz, Lucía Ivonne Hernández-Martínez, Federico Sánchez-Bringas
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Classification of Calabi Hypersurfaces in $${{\mathbb {R}}}^{n+1}$$ with Parallel Fubini-Pick Tensor J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-09 Miaoxin Lei, Ruiwei Xu
The classifications of locally strongly convex equiaffine hypersurfaces (resp. centroaffine hypersurfaces) with parallel Fubini–Pick tensor with respect to the Levi-Civita connection of the Blaschke–Berwald affine metric (resp. centroaffine metric) have been completed in the last decades. In this paper we define generalized Calabi products in Calabi geometry and prove decomposition theorems in terms
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Remarks on the Possible Blow-Up Conditions via One Velocity Component for the 3D Navier–Stokes Equations J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-09 Zhengguang Guo, Chol-Jun O
In this paper, we study some blow-up conditions via one velocity component for the 3D incompressible Navier–Stokes equations in the framework of scaling invariant anisotropic Besov spaces. In particular, we prove that if one component of the velocity remains small enough in the space \(\dot{H}^{\frac{1}{2}}\), then there is no blow-up. This result improves the previous ones by Chemin et al. (Commun
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Isoparametric Functions and Solutions of Yamabe Type Equations on Manifolds with Boundary J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-08 Guillermo Henry, Juan Zuccotti
Let (M, g) be a compact Riemannian manifold with non-empty boundary. Provided that f is an isoparametric function of (M, g), we prove existence results for positive solutions of the Yamabe equation that are constant along the level sets of f. If (M, g) has positive constant scalar curvature, minimal boundary and admits an isoparametric function we also prove multiplicity results for positive solutions
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Positive Ricci Curvature and the Length of a Shortest Periodic Geodesic J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-08 Regina Rotman
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Integrability and Symmetry of Positive Integrable Solutions for Weighted Wolff-Type Integral Systems J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-08 Yan Bai, Zexin Zhang, Zhitao Zhang
In this paper, we are concerned with the following weighted integral system involving Wolff potential: $$\begin{aligned} \left\{ \begin{array}{lll} u(x)=R_1(x)W_{\beta ,\gamma }\left( \frac{v^q}{|y|^\sigma }\right) (x),\quad &{} u(x)>0,\quad &{} x\in \mathbb {R}^N,\\ v(x)=R_2(x)W_{\beta ,\gamma }\left( \frac{u^p}{|y|^\sigma }\right) (x),\quad &{} v(x)>0,\quad &{} x\in \mathbb {R}^N, \end{array}\right
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Regularly Oscillating Mappings Between Metric Spaces and a Theorem of Hardy and Littlewood J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-08 Marijan Marković
This paper is motivated by the classical theorem due to Hardy and Littlewood which concerns analytic mappings on the unit disk and relates the growth of the derivative with the Hölder continuity. We obtain a version of this result in a very general setting—for regularly oscillating mappings on a metric space equipped with a weight, which is a continuous and positive function, with values in another
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Asymptotic Uniqueness of Minimizers for Hartree Type Equations with Fractional Laplacian J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-08 Lintao Liu, Kaimin Teng, Shuai Yuan
We study the concentration and uniqueness of standing waves associated with the constraint minimization problems for the nonlinear Hartree type equations with homogeneous potentials and fractional Laplacian. This class of equations is an effective model to describe the fractional quantum mechanics with a convolution perturbation. By making full use of the Bessel kernel and adopting the iterative process
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Kähler Geometry of Scalar Flat Metrics on Line Bundles Over Polarized Kähler–Einstein Manifolds J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 Simone Cristofori, Michela Zedda
Abstract In view of a better understanding of the geometry of scalar flat Kähler metrics, this paper studies two families of scalar flat Kähler metrics constructed by Hwang and Singer (Trans Am Math Soc 354(6):2285–2325, 2002) on \(\mathbb {C}^{n+1}\) and on \({\mathcal {O}}(-k)\) . For the metrics in both the families, we prove the existence of an asymptotic expansion for their \(\epsilon \) -functions
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Modulus Characterizations of Bilipschitz Mappings J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 Qingshan Zhou, Zhiqiang Yang, Antti Rasila, Yuehui He
Abstract In this paper, we establish six necessary and sufficient conditions for a homeomorphism of \(\mathbb {R}^n\) onto itself to be strongly quasisymmetric. These conditions are quantitative in terms of conformal moduli of disjoint continua as well as the geometric modulus, which was recently introduced by Tukia and Väisälä. Note that all of them are equivalent to bilipschitz continuity with parameters
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Pluri-Potential Theory, Submersions and Calibrations J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 Tommaso Pacini
Abstract We present a systematic collection of results concerning interactions between convex, subharmonic and pluri-subharmonic functions on pairs of manifolds related by a Riemannian submersion. Our results are modelled on those known in the classical complex-analytic context and represent another step in the recent Harvey–Lawson pluri-potential theory for calibrated manifolds. In particular, we
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A Bilinear Sparse Domination for the Maximal Singular Integral Operators with Rough Kernels J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 Xiangxing Tao, Guoen Hu
Abstract Let \(\Omega \) be homogeneous of degree zero, integrable on \(S^{d-1}\) and have mean value zero, \(T_{\Omega }\) be the homogeneous singular integral operator with kernel \(\frac{\Omega (x)}{|x|^d}\) and \(T_{\Omega }^*\) be the maximal operator associated to \(T_{\Omega }\) . In this paper, the authors prove that if \(\Omega \in L^{\infty }(S^{d-1})\) , then for all \(r\in (1,\,\infty )\)
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Geometric and Analytic Properties Associated With Extension Operators J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 Jianfei Wang, Taishun Liu, Yanhui Zhang
Abstract The first aim is to prove that the Roper-Suffridge extension operator preserves \(\varepsilon \) -starlike property on general domains given by convex functions. The second is to construct the generalized Roper-Suffridge extension operator on Reinhard domains $$\begin{aligned} \Omega _{p_{1},p_{2},\cdots ,p_{n}}=\{(z_1,\ldots ,z_{n})\in {\mathbb {C}}^{n}:\sum \limits _{j=1}^{n}|z_{j}|^{p_{j}}<1\}
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Rigidity Theorem for Integral Pinched Static Manifolds and Related Critical Spaces J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 H. Baltazar, C. Queiroz
Abstract The aim of this paper is to investigate positive static triples, critical metrics of volume functional, and critical metrics of the total scalar curvature functional satisfying a \(L^{n/2}\) -pinching condition.
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Mean-Dispersion Principles and the Wigner Transform J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 Chiara Boiti, David Jornet, Alessandro Oliaro
Abstract Given a function \(f\in L^2(\mathbb {R})\) , we consider means and variances associated to f and its Fourier transform \(\hat{f}\) , and explore their relations with the Wigner transform W(f), obtaining, as particular cases, a simple new proof of Shapiro’s mean-dispersion principle, as well as a stronger result due to Jaming and Powell. Uncertainty principles for orthonormal sequences in \(L^2(\mathbb
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The Dual Minkowski Problem for p-Capacity J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 Lewen Ji
Abstract A new family of geometric Borel measures on the unit sphere is introduced, which extends the \(L_0\) p-capacitary measures proposed by Zou and Xiong (J Differ Geom 116:555–596, 2020). In this paper, we consider the existence of the solution to the p-capacitary dual Minkowski problem when \(1
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Uniformly Perfect Sets, Hausdorff Dimension, and Conformal Capacity J. Geom. Anal. (IF 1.1) Pub Date : 2024-04-06 Oona Rainio, Toshiyuki Sugawa, Matti Vuorinen
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The Burns-Krantz Type Rigidity for Domains With Corners J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-25 Feng Rong
First, we extend the Burns-Krantz rigidity for the unit disk to domains with corners. Then, we prove the Burns-Krantz type rigidity for some fibered domains over these one-dimensional domains with corners.
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The Covariance Metric in the Blaschke Locus J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-25 Xian Dai, Nikolas Eptaminitakis
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Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-25 Jiguang Bao, Zixiao Liu, Cong Wang
In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the
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Killing Fields on Compact Pseudo-Kähler Manifolds J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-25 Andrzej Derdzinski, Ivo Terek
We show that a Killing field on a compact pseudo-Kähler ddbar manifold is necessarily (real) holomorphic. Our argument works without the ddbar assumption in real dimension four. The claim about holomorphicity of Killing fields on compact pseudo-Kähler manifolds appears in a 2012 paper by Yamada, and in an appendix we provide a detailed explanation of why we believe that Yamada’s argument is incomplete
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Some Functional Inequalities and Their Applications on Finsler Measure Spaces J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-15 Xinyue Cheng, Yalu Feng
We study functional and geometric inequalities on complete Finsler measure spaces with the weighted Ricci curvature \(\textrm{Ric}_\infty \) bounded below. We first obtain some local uniform Poincaré inequalities and Sobolev inequalities. Then, we prove a mean value inequality for nonnegative subsolutions of elliptic equations. Further, we derive local and global Harnack inequalities for positive harmonic
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Lipschitz-Volume Rigidity and Sobolev Coarea Inequality for Metric Surfaces J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-15 Damaris Meier, Dimitrios Ntalampekos
We prove that every 1-Lipschitz map from a closed metric surface onto a closed Riemannian surface that has the same area is an isometry. If we replace the target space with a non-smooth surface, then the statement is not true and we study the regularity properties of such a map under different geometric assumptions. Our proof relies on a coarea inequality for continuous Sobolev functions on metric
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Positive Intermediate Ricci Curvature with Maximal Symmetry Rank J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-15 Lee Kennard, Lawrence Mouillé
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Nonlinear Dirac Equation on Compact Spin Manifold with Chirality Boundary Condition J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-15 Yanyun Wen, Peihao Zhao
In this paper, we study the following nonlinear boundary value problem $$\begin{aligned} \left\{ \begin{array}{ll} D\psi -a(x)\psi =f(x,\psi )+\epsilon h(x,\psi )&{} \quad \hbox {on }M \\ B_{CHI}\psi =0 &{} \quad \hbox {on }\partial M \end{array} \right. \end{aligned}$$(D) where M is a compact Riemannian spin manifold of dimension\(m\ge 2\) and the boundary \(\partial M\) has non-negative mean curvature
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Small-Constant Uniform Rectifiability J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-14 Cole Jeznach
We provide several equivalent characterizations of locally flat, d-Ahlfors regular, uniformly rectifiable sets E in \({\mathbb {R}}^n\) with density close to 1 for any dimension \(d \in {\mathbb {N}}\), \(1 \le d < n\). In particular, we show that when E is Reifenberg flat with small constant and has Ahlfors regularity constant close to 1, then the Tolsa \(\alpha \) coefficients associated to E satisfy
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Toward Weighted Lorentz–Sobolev Capacities from Caffarelli–Silvestre Extensions J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-12 Xing Fu, Jie Xiao, Qi Xiong
Abstract Getting inspired by the Caffarelli–Silvestre extensions, this paper investigates the weighted Lorentz–Sobolev capacities and their capacitary strong inequalities with applications to the Sobolev-type embeddings. Consequently, the weighted Lebesgue-Sobolev capacities and their applications to a functional inequality problem and the existence-regularity of solutions to the prototype p-Laplace
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Direct Minimization of the Canham–Helfrich Energy on Generalized Gauss Graphs J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-09 Anna Kubin, Luca Lussardi, Marco Morandotti
The existence of minimizers of the Canham–Helfrich functional in the setting of generalized Gauss graphs is proved. As a first step, the Canham–Helfrich functional, usually defined on regular surfaces, is extended to generalized Gauss graphs, then lower semicontinuity and compactness are proved under a suitable condition on the bending constants ensuring coerciveness; the minimization follows by the
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Weighted K-Stability for a Class of Non-compact Toric Fibrations J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-09 Charles Cifarelli
We study the weighted constant scalar curvature, a modified scalar curvature introduced by Lahdili (Proc Lond Math Soc 119(4):1065–1114, 2019) depending on weight functions \((v, \, w)\), on non-compact semisimple principal toric fibrations. The latter notion is a generalization of the Calabi Ansatz originally defined by Apostolov et al. (J Differ Geom 68(2):277–345, 2004). This setup turns out to
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Lower Bounds for the First Eigenvalue of the p-Laplacian on Quaternionic Kähler Manifolds J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-09 Kui Wang, Shaoheng Zhang
We study the first nonzero eigenvalues for the p-Laplacian on quaternionic Kähler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the p-Laplacian on compact quaternionic Kähler manifolds. Our second result is a lower bound for the first Dirichlet eigenvalue of the p-Laplacian on compact quaternionic Kähler manifolds with smooth boundary. Our results
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The Heterotic-Ricci Flow and Its Three-Dimensional Solitons J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-09 Andrei Moroianu, Ángel J. Murcia, C. S. Shahbazi
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Ancient Solutions of Ricci Flow with Type I Curvature Growth J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-08 Stephen Lynch, Andoni Royo Abrego
Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete
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$$L^p_{loc}$$ Positivity Preservation and Liouville-Type Theorems J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-02 Andrea Bisterzo, Alberto Farina, Stefano Pigola
On a complete Riemannian manifold (M, g), we consider \(L^{p}_{loc}\) distributional solutions of the differential inequality \(-\Delta u + \lambda u \ge 0\) with \(\lambda >0\) a locally bounded function that may decay to 0 at infinity. Under suitable growth conditions on the \(L^{p}\) norm of u over geodesic balls, we obtain that any such solution must be nonnegative. This is a kind of generalized
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Pseudo-Kähler Geometry of Properly Convex Projective Structures on the torus J. Geom. Anal. (IF 1.1) Pub Date : 2024-03-01 Nicholas Rungi, Andrea Tamburelli
In this paper we prove the existence of a pseudo-Kähler structure on the deformation space \({\mathcal {B}}_0(T^2)\) of properly convex \({\mathbb {R}}{\mathbb {P}}^2\)-structures over the torus. In particular, the pseudo-Riemannian metric and the symplectic form are compatible with the complex structure inherited from the identification of \({\mathcal {B}}_0(T^2)\) with the complement of the zero
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p-Carleson Measures in the Quaternionic Unit Ball with Applications to Slice Campanato and $$Q_p$$ Spaces J. Geom. Anal. (IF 1.1) Pub Date : 2024-02-28 Cheng Yuan
The p-Carleson measure in the unit ball of quaternions is introduced in terms of the symmetric box. When \(p=1\) or \(p=2\), the p-Carleson measure becomes the Carleson measure for the Hardy or Bergman spaces, respectively. A criterion for a measure to be a p-Carleson measure is provided in terms of slice Cauchy kernels. Bergman type integral operators are shown to preserve the p-Carleson measure in
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Equivariant Solutions to the Optimal Partition Problem for the Prescribed Q-Curvature Equation J. Geom. Anal. (IF 1.1) Pub Date : 2024-02-23 Juan Carlos Fernández, Oscar Palmas, Jonatán Torres Orozco
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Parabolicity of Invariant Surfaces J. Geom. Anal. (IF 1.1) Pub Date : 2024-02-22
Abstract We present a clear and practical way to characterize the parabolicity of a complete immersed surface that is invariant with respect to a Killing vector field of the ambient space.
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Restricted Mean Value Property on Riemannian manifolds J. Geom. Anal. (IF 1.1) Pub Date : 2024-02-22 Kingshook Biswas, Utsav Dewan
A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP). While this has so far been studied mainly for domains in \(\mathbb {R}^n\), we consider this problem in the general setting of domains in Riemannian manifolds, and obtain results generalizing classical results of Fenton. We also obtain a result for complete, simply connected Riemannian
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Pluriclosed Flow and Hermitian-Symplectic Structures J. Geom. Anal. (IF 1.1) Pub Date : 2024-02-22 Yanan Ye
We observe that pluriclosed flow preserves Hermitian-symplectic structures. And we extend pluriclosed flow to a flow of Hermitian-symplectic forms by adding an extra evolution equation, which is determined by the (2, 0)-part of Bismut–Ricci form. Moreover, we obtain a topological obstruction to the existence of global solutions in every dimension.
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Some Functional Properties on Cartan–Hadamard Manifolds of Very Negative Curvature J. Geom. Anal. (IF 1.1) Pub Date : 2024-02-20 Ludovico Marini, Giona Veronelli
In this paper, we consider Cartan–Hadamard manifolds (i.e., simply connected, complete, of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold on manifolds of bounded curvature, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called
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Strict Monotonicity of the First q-Eigenvalue of the Fractional p-Laplace Operator Over Annuli J. Geom. Anal. (IF 1.1) Pub Date : 2024-02-08 K. Ashok Kumar, Nirjan Biswas
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A Hessian-Dependent Functional With Free Boundaries and Applications to Mean-Field Games J. Geom. Anal. (IF 1.1) Pub Date : 2024-02-08 Julio C. Correa, Edgard A. Pimentel
We study a Hessian-dependent functional driven by a fully nonlinear operator. The associated Euler-Lagrange equation is a fully nonlinear mean-field game with free boundaries. Our findings include the existence of solutions to the mean-field game, together with Hölder continuity of the value function and improved integrability of the density. In addition, we prove the reduced free boundary is a set
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Dual Integrable Representations on Locally Compact Groups J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-30 Hrvoje Šikić, Ivana Slamić
Studies of various reproducing function systems emphasized the role of translations and the Fourier periodization function. These influenced the development of the concept of dual integrable representations, a large and important class of unitary representations on LCA groups. The key ingredient is the bracket function that enables the explicit description of corresponding cyclic spaces. Since its
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Classification of Solutions to Several Semi-linear Polyharmonic Equations and Fractional Equations J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-25 Zhuoran Du, Zhenping Feng, Yuan Li
We consider the following semi-linear equations $$\begin{aligned} (-\Delta )^pu=u^\gamma _+ ~~ \text{ in } {{\mathbb {R}}^n}, \end{aligned}$$ where \(\gamma \in (1,\frac{n+2p}{n-2p})\), \(n>2p>0\), \(u_+=\max \{u,0\}\), and \(2\le p\in {\mathbb {N}}\) or \(p\in (0,1)\). Subject to the integral constraint $$\begin{aligned} u_+^\gamma \in L^1({\mathbb {R}}^n), \end{aligned}$$ we obtain the classification
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A Sharp Sobolev Principle on the Graphic Submanifolds of $${\mathbb {R}}^{n+m}$$ J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-25 Jie Xiao, Fanheng Xu
This paper shows such a sharp Sobolev principle that if \((\Sigma ,g)\) is a compact n-dimensional graphic submanifold of \({\mathbb {R}}^{n+m}\), \(G=|\text {det}g|\) is the absolute value of the determinant of g, \(|B^n|\) is the volume of the open unit ball \(B^n\) in \({\mathbb {R}}^n\), and f is a positive smooth function on \(\Sigma \), then $$\begin{aligned} \left( \frac{\int _\Sigma |\nabla
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Legendrian Mean Curvature Flow in $$\eta $$ -Einstein Sasakian Manifolds J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-25 Shu-Cheng Chang, Yingbo Han, Chin-Tung Wu
Abstract Recently, there are a great deal of work done which connects the Legendrian isotopic problem with contact invariants. The isotopic problem of Legendre curve in a contact 3-manifold was studied via the Legendrian curve shortening flow which was introduced and studied by K. Smoczyk. On the other hand, in the SYZ Conjecture, one can model a special Lagrangian singularity locally as the special
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Existence and Concentration of Solutions to a Choquard Equation Involving Fractional p-Laplace via Penalization Method J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-25 Xin Zhang, Xueqi Sun, Sihua Liang, Van Thin Nguyen
In this paper, we study the Choquard equation involving \(\frac{N}{s}\)-fractional Laplace as follows: $$\begin{aligned} \varepsilon ^{ps}(-\Delta )_{p}^{s}u+V(x)|u|^{p-2}u= \varepsilon ^{\mu -N}\left[ \dfrac{1}{|x|^{\mu }}*F(u)\right] f(u)\;\text {in}\; \mathbb {R}^{N}, \end{aligned}$$ where \(\varepsilon \) is a positive parameter, \(N=ps, s\in (0,1), 0<\mu
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On Highly Degenerate CR Maps of Spheres J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-19 Giuseppe della Sala, Bernhard Lamel, Michael Reiter, Duong Ngoc Son
For \(N \ge 4\) we classify the \((N-3)\)-degenerate smooth CR maps of the three-dimensional unit sphere into the \((2N-1)\)-dimensional unit sphere. Each of these maps has image being contained in a five-dimensional complex-linear space and is of degree at most two, or equivalent to one of the four maps into the five-dimensional sphere classified by Faran. As a byproduct of our classification we obtain
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Global Solutions of 2-D Cubic Dirac Equation with Non-compactly Supported Data J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-19 Qian Zhang
We are interested in the cubic Dirac equation in two space dimensions. We establish the small data global existence and sharp time decay results for general cubic nonlinearities without additional structure. We also prove the scattering of the Dirac equation for certain classes of nonlinearities. In all the above results we do not require the initial data to have compact support.
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Gaps in the Support of Canonical Currents on Projective K3 Surfaces J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-19 Simion Filip, Valentino Tosatti
We construct examples of canonical closed positive currents on projective K3 surfaces that are not fully supported on the complex points. The currents are the unique positive representatives in their cohomology classes and have vanishing self-intersection. The only previously known such examples were due to McMullen on nonprojective K3 surfaces and were constructed using positive entropy automorphisms
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Well-Posedness of the Kadomtsev–Petviashvili-II in the Negative Sobolev Space with Respect to y Direction J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-19 Zhaohui Huo
In this paper, we first establish some new dyadic bilinear estimates of KP-II equation. Then following some ideas in [7], we consider the Cauchy problem of the 2-D KP-II equation in negative Sobolev space with respect to y direction $$\begin{aligned} \partial _t u + \partial _{xxx}u+\partial _{x}^{-1}(\partial _{yy} )u +\partial _x (u^2) =0, \ (x,y,t) \in {\mathbb {R}}^3. \end{aligned}$$ It follows
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Subsets of Positive and Finite $$\Psi _t$$ -Hausdorff Measures and Applications J. Geom. Anal. (IF 1.1) Pub Date : 2024-01-19 Bilel Selmi
Working with sets of infinite \(\Psi _t\)-Hausdorff measures can be cumbersome, so simplifying them to sets of positive finite general fractal measures proves to be highly beneficial. This paper aims to demonstrate that the \(\Psi _t\)-Hausdorff measures adhere to the property of being a subset of positive and finite measures. Our main result is then applied to establish that the general fractal function