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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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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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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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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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Rytis Juršėnas 《Mathematische Nachrichten》2023,296(8):3411-3448
Let be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space . Let be the Weyl family corresponding to . We cope with two main topics. First, since need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation , for some , becomes a nontrivial task. Regarding as the (Shmul'yan) transform of induced by Γ, we give conditions for the equality in to hold and we compute the adjoint . As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for is nonempty. Based on the criterion for the closeness of , we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family corresponding to a boundary relation Γ for is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair with its Weyl family . The transformation scheme is either or with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between and ; the formula for , for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple with and (second scheme, Hilbert space case). 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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Inspired by some iterative algorithms useful for proving the real analyticity (or the Gevrey regularity) of a solution of a linear partial differential equation with real-analytic coefficients, we consider the following question. Given a smooth function defined on and given an increasing divergent sequence of positive integers such that the derivative of order of f has a growth of the type , when can we deduce that f is a function in the Denjoy–Carleman class ? We provide a positive result and show that a suitable condition on the gaps between the terms of the sequence is needed. 相似文献
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Giovany M. Figueiredo Marcelo Montenegro Matheus F. Stapenhorst 《Mathematische Nachrichten》2023,296(10):4569-4609
We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, in with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional corresponding to the perturbed equation , where is well defined at 0 and approximates . We show that has a critical point in , which converges to a legitimate nontrivial nonnegative solution of the original problem as . We also investigate the problem with replaced by , when the parameter is sufficiently large. 相似文献
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We show that if a densely defined closable operator A is such that the resolvent set of A2 is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial . We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that is normal for some , then T is normal. Hence a closed subnormal operator T such that is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that and are self-adjoint for some coprime numbers p and q, then A must be self-adjoint. 相似文献
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María Inés Gareis Federico Dario Kovac Fabián Eduardo Levis 《Mathematische Nachrichten》2023,296(8):3328-3343
In this paper, we consider the best polynomial approximation operator defined on an Orlicz–Lorentz space , and its extension to , where w is a non-negative continuous weight function and is the derivative of ϕ, which is not required to be an Orlicz function. Our work generalizes a recent result in this field on an Orlicz–Lorentz space generated by an Orlicz function. In addition, we establish some properties and estimates for any extended best polynomial approximation. 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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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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A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence so that as for every weakly null sequence in X; X has property if every weak* null sequence in admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces. 相似文献