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Let be either a simply connected space form or a rank-one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of , which are those whose principal curvatures and angle function satisfy a relation , being W a differentiable function which is symmetric with respect to . When on the positive cone of , a strictly convex Weingarten hypersurface determined by W is said to be elliptic. We show that, for a certain class of Weingarten functions W, there exist rotational strictly convex Weingarten hypersurfaces of which are either topological spheres or entire graphs over M. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either or is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of and which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of and . 相似文献
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Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood–Paley square function for Dunkl heat flows in is introduced by employing the full “gradient” induced by the corresponding carré du champ operator and then the boundedness is studied for all . For , we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the boundedness, while for , we restrict to a particular case when the corresponding Weyl group is isomorphic to and apply a probabilistic method to prove the boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry–Emery, which may be of independent interest, plays a crucial role. The results are dimension-free. 相似文献
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The main aim of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary and under the harmonic Weyl tensor condition. In particular, we prove that a critical metric with a harmonic Weyl tensor on a simply connected compact manifold with the boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form , , and . To this end, we first conclude the classification of such critical metrics under the Bach-flat assumption and then we prove that both geometric conditions are equivalent in this situation. 相似文献
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The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves and in terms of the finiteness and the vanishing properties of the invariants of and , and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial is a power of , and we characterize when in terms of the defining equations of and . Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves. 相似文献
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Vicente Lorenzo 《Mathematische Nachrichten》2023,296(6):2503-2512
In this note, the geography of minimal surfaces of general type admitting -actions is studied. More precisely, it is shown that Gieseker's moduli space contains surfaces admitting a -action for every admissible pair such that or . The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification of Gieseker's moduli space for every admissible pair such that . 相似文献
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Roger Bielawski 《Mathematische Nachrichten》2023,296(1):122-129
We show that -invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in correspond to algebraic curves C of genus , equipped with a flat projection of degree k, and an antiholomorphic involution covering the antipodal map on . 相似文献
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Aurélio Menegon 《Mathematische Nachrichten》2023,296(8):3481-3491
Let (X, 0) be the germ of either a subanalytic set or a complex analytic space , and let be a -analytic map-germ, with or , respectively. When , there is a well-known topological locally trivial fibration associated with f, called the Milnor–Lê fibration of f, which is one of the main pillars in the study of singularities of maps and spaces. However, when that is not always the case. In this paper, we give conditions which guarantee that the image of f is well-defined as a set-germ, and that f admits a Milnor–Lê fibration. We also give conditions for f to have the Thom property. Finally, we apply our results to mixed function-germs of type on a complex analytic surface with arbitrary singularity. 相似文献
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In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric , and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both and , we determine them for , and give a classification theorem when they are Clifford solutions. 相似文献
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We study the conormal sheaves and singular schemes of one-dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ-stable whenever the tangent bundle is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on and on a smooth quadric hypersurface . Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q3. 相似文献
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We obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the metric of the space of quasi-continuous functions . We also showed that for , , , , the estimate of the corresponding asymptotic characteristic is exact in order. 相似文献
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We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in , , with respect to subordinate killed Brownian motions via -stable subordinators and establish the asymptotic behavior of the spectral heat content as for the full range of . Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when , where is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem. 相似文献
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We investigate the topological structures of Galois covers of surfaces of minimal degree (i.e., degree n) in . We prove that for , the Galois covers of any surfaces of minimal degree are simply-connected surfaces of general type. 相似文献
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We show that if is a one-parameter continuous semigroup of nonexpansive mappings acting on a complete locally compact geodesic space that satisfies some geometric properties, then there exists such that S converge uniformly on bounded sets of Y to ξ. In particular, our result applies to strictly convex bounded domains in or with respect to a large class of metrics including Hilbert's and Kobayashi's metrics. 相似文献
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Paulo Cesar Carrião Olímpio Hiroshi Miyagaki André Vicente 《Mathematische Nachrichten》2023,296(1):130-151
In this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space . The space is the unit disc of endowed with the Riemannian metric g given by , where and , if and , if . Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis–Marcus, which allows us to overcome the difficulty involving the singularities. 相似文献
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In order to improve the classical Bohr inequality, we explain some refined versions for a quasi-subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk . Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form , where , the analytic part h is bounded by 1 and that in and for some . 相似文献