Authors:Diego Conti; Federico A. Rossi Pages: 467 - 501 Abstract: We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi–Civita connection. The formula uses the intrinsic torsion of an underlying \(\mathrm {SL}(n,\mathbb {R})\) -structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parakähler version of the Goldberg conjecture and obtain the first compact examples of a non-flat, Ricci-flat nearly parakähler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parakähler metrics. PubDate: 2018-06-01 DOI: 10.1007/s10455-017-9584-y Issue No:Vol. 53, No. 4 (2018)
Authors:Oscar Macia; Yasuyuki Nagatomo Pages: 503 - 520 Abstract: The present article studies the class of Einstein–Hermitian harmonic maps of constant Kähler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image and gauge equivalence using the language of vector bundles and representation theory. It is shown that the dimension of the moduli spaces is independent of the Einstein–Hermitian constant and rigidity of the associated real standard, and totally real maps are examined. Finally, certain classical results concerning embeddings of two-dimensional spheres into spheres are rephrased and derived in our formalism. PubDate: 2018-06-01 DOI: 10.1007/s10455-017-9585-x Issue No:Vol. 53, No. 4 (2018)
Authors:Rafael López Pages: 521 - 541 Abstract: We classify all singular minimal surfaces in Euclidean space that are invariant by a uniparametric group of translations and rotations. PubDate: 2018-06-01 DOI: 10.1007/s10455-017-9586-9 Issue No:Vol. 53, No. 4 (2018)
Authors:Ignacio Bajo; Esperanza Sanmartín Pages: 543 - 559 Abstract: Hyper-para-Kähler structures on Lie algebras where the complex structure is abelian are studied. We show that there is a one-to-one correspondence between such hyper-para-Kähler Lie algebras and complex commutative (hence, associative) symplectic left-symmetric algebras admitting a semilinear map \(K_s\) verifying certain algebraic properties. Such equivalence allows us to give a complete classification, up to holomorphic isomorphism, of pairs \(({\mathfrak g},J)\) of 8-dimensional Lie algebras endowed with abelian complex structures which admit hyper-para-Kähler structures. PubDate: 2018-06-01 DOI: 10.1007/s10455-017-9587-8 Issue No:Vol. 53, No. 4 (2018)
Authors:Hilário Alencar; Adina Rocha Pages: 561 - 581 Abstract: In this paper, we study stability properties of hypersurfaces with constant weighted mean curvature (CWMC) in gradient Ricci solitons. The CWMC hypersurfaces generalize the f-minimal hypersurfaces and appear naturally in the isoperimetric problems in smooth metric measure spaces. We obtain a result about the relationship between the properness and extrinsic volume growth under the assumption of a limitation for the weighted mean curvature of the immersion. Moreover, we estimate Morse index for CWMC hypersurfaces in terms of the dimension of the space of parallel vector fields restricted to hypersurface. PubDate: 2018-06-01 DOI: 10.1007/s10455-017-9588-7 Issue No:Vol. 53, No. 4 (2018)
Authors:Hui Ma; Andrey E. Mironov; Dafeng Zuo Pages: 583 - 595 Abstract: A two-dimensional periodic Schrödingier operator is associated with every Lagrangian torus in the complex projective plane \({\mathbb C}P^2\) . Using this operator, we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional \(W^{-}\) introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel–Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e., preserves the value of the energy functional. In particular, the deformations generated by Novikov–Veselov equations preserve the area of minimal Lagrangian tori. PubDate: 2018-06-01 DOI: 10.1007/s10455-017-9589-6 Issue No:Vol. 53, No. 4 (2018)
Authors:Renato G. Bettiol; Ricardo A. E. Mendes Pages: 287 - 309 Abstract: We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive-definiteness of the curvature operator and positivity of sectional curvature, and stems from the work of Thorpe (J Differ Geom 5:113–125, 1971; Erratum. J Differ Geom 11:315, 1976). We prove that this condition is preserved under Riemannian submersions and Cheeger deformations and that most compact homogeneous spaces with positive sectional curvature satisfy it. PubDate: 2018-04-01 DOI: 10.1007/s10455-017-9578-9 Issue No:Vol. 53, No. 3 (2018)
Authors:Liviu Ornea; Vladimir Slesar Pages: 311 - 329 Abstract: In this paper we investigate the spectral sequence associated with a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. We show that if the foliation is quasi-regular the lower bound is realized. In the final part of the paper we discuss two examples. PubDate: 2018-04-01 DOI: 10.1007/s10455-017-9579-8 Issue No:Vol. 53, No. 3 (2018)
Authors:Ivan Minchev; Jan Slovák Pages: 331 - 375 Abstract: Following the Cartan’s original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete system of differential invariants for two quaternionic contact manifolds to be locally diffeomorphic. This includes an explicit construction of the corresponding Cartan geometry and detailed information on all curvature components. PubDate: 2018-04-01 DOI: 10.1007/s10455-017-9580-2 Issue No:Vol. 53, No. 3 (2018)
Authors:Zhenxiao Xie; Tongzhu Li; Xiang Ma; Changping Wang Pages: 377 - 403 Abstract: Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. As conformal invariant objects, they are suitable to study in the framework of Möbius geometry. This paper continues our previous work in this program, showing that Wintgen ideal submanifolds can be divided into three classes: the reducible ones, the irreducible minimal ones in space forms (up to Möbius transformations), and the generic (irreducible) ones. The reducible Wintgen ideal submanifolds have a specific low-dimensional integrable distribution, which allows us to get the most general reduction theorem, saying that they are Möbius equivalent to cones, cylinders, or rotational surfaces generated by minimal Wintgen ideal submanifolds in lower-dimensional space forms. PubDate: 2018-04-01 DOI: 10.1007/s10455-017-9581-1 Issue No:Vol. 53, No. 3 (2018)
Authors:Daniel J. F. Fox Pages: 405 - 443 Abstract: The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood–Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra. PubDate: 2018-04-01 DOI: 10.1007/s10455-017-9582-0 Issue No:Vol. 53, No. 3 (2018)
Authors:Daniel Grady; Hisham Sati Pages: 445 - 466 Abstract: Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main properties of the new theory and illustrate its use with examples and applications. Given how versatile Deligne cohomology has proven to be, we believe that this explicit and utilizable treatment of its twisted version will be useful. PubDate: 2018-04-01 DOI: 10.1007/s10455-017-9583-z Issue No:Vol. 53, No. 3 (2018)
Abstract: We define virtual immersions, as a generalization of isometric immersions in a pseudo-Riemannian vector space. We show that virtual immersions possess a second fundamental form, which is in general not symmetric. We prove that a manifold admits a virtual immersion with skew-symmetric second fundamental form, if and only if it is a symmetric space, and in this case the virtual immersion is essentially unique. PubDate: 2018-05-30 DOI: 10.1007/s10455-018-9617-1
Authors:Fabio Podestà; Alberto Raffero Abstract: We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result of Wolf and Gray allowing one to show the nonexistence of compact non-flat examples. In the noncompact setting, we classify such manifolds under the assumption that G is semisimple. Moreover, in each case, we describe all invariant symplectic half-flat SU(3)-structures up to isomorphism, showing that the Ricci tensor is always Hermitian with respect to the induced almost complex structure. This property of the Ricci tensor is characterized in the general case. PubDate: 2018-05-24 DOI: 10.1007/s10455-018-9615-3
Authors:Jia-Yong Wu Abstract: In this paper, we prove a new Myers’ type diameter estimate on a complete connected Reimannian manifold which admits a bounded vector field such that the Bakry–Émery Ricci tensor has a positive lower bound. The result is sharper than previous Myers’ type results. The proof uses the generalized mean curvature comparison applied to the excess function instead of the classical second variation of geodesics. PubDate: 2018-05-14 DOI: 10.1007/s10455-018-9613-5
Authors:Donato Cianci; Alexandre Girouard Abstract: In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectral gap arbitrarily large using conformal deformations which are localized on domains of small measure, as long as the support of the deformations contains and connects each component of the boundary. PubDate: 2018-05-08 DOI: 10.1007/s10455-018-9612-6
Authors:Jeffrey L. Jauregui Abstract: The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities. PubDate: 2018-04-26 DOI: 10.1007/s10455-018-9611-7
Authors:Serdar Altuntas Abstract: We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in supercritical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps. PubDate: 2018-04-16 DOI: 10.1007/s10455-018-9610-8
Authors:Hwajeong Kim Abstract: A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \({\mathcal E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \({\mathcal E}\) . We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \({\mathcal E}\) is non-degenerated. PubDate: 2018-04-11 DOI: 10.1007/s10455-018-9601-9
Authors:Peijun Wang; Xiaoli Chao; Yilong Wu; Yusha lv Abstract: In the present note, the geometric structures and topological properties of harmonic p-forms on a complete noncompact submanifold \(M^{n}(n\ge 4)\) immersed in Hadamard manifold \(N^{n+m}\) are discussed, where \(M^{n}\) and \(N^{n+m}\) are assumed to have flat normal bundle and pure curvature tensor, respectively. Firstly, under the assumption that \(M^{n}\) satisfies the \((\mathcal {P}_\rho )\) property (i.e., the weighted Poincaré inequality holds on \(M^{n}\) ) and the \((p,n-p)\) -curvature of \(N^{n+m}\) is not less than a given negative constant, using Moser iteration, the space of all \(L^{2}\) harmonic p-forms on \(M^{n}\) is proven to have finite dimensions if \(M^{n}\) has finite total curvature. Furthermore, if the total curvature is small enough or \(M^{n}\) has at most Euclidean volume growth, two vanishing theorems are, respectively, established for harmonic p-forms. Note that the two vanishing theorems extend several previous results obtained by H. Z. Lin. PubDate: 2018-04-05 DOI: 10.1007/s10455-018-9609-1
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