Authors:Jesús A. Álvarez López; Manuel Calaza; Carlos Franco Abstract: Abstract Let M be a stratum of a compact stratified space A. It is equipped with a general adapted metric g, which is slightly more general than the adapted metrics of Nagase and Brasselet–Hector–Saralegi. In particular, g has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then, g is called good. We consider the maximum/minimum ideal boundary condition, \(d_{\mathrm{max/min}}\) , of the compactly supported de Rham complex on M, in the sense of Brüning–Lesch. Let \(H^*_{\mathrm{max/min}}(M)\) and \(\Delta _{\mathrm{max/min}}\) denote the cohomology and Laplacian of \(d_{\mathrm{max/min}}\) . The first main theorem states that \(\Delta _{\mathrm{max/min}}\) has a discrete spectrum satisfying a weak form of the Weyl’s asymptotic formula. The second main theorem is a version of Morse inequalities using \(H_{\mathrm{max/min}}^*(M)\) and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for \(d_{\mathrm{max/min}}\) of the Witten’s perturbation of the de Rham complex. Another ingredient is certain perturbation of the Dunkl harmonic oscillator previously studied by the authors using classical perturbation theory. The condition on g to be good is general enough in the following sense. Assume that A is a stratified pseudomanifold, and consider its intersection homology \(I^{\bar{p}}H_*(A)\) with perversity \(\bar{p}\) ; in particular, the lower and upper middle perversities are denoted by \(\bar{m}\) and \(\bar{n}\) , respectively. Then, for any perversity \(\bar{p}\le \bar{m}\) , there is an associated good adapted metric on M satisfying the Nagase isomorphism \(H^r_{\mathrm{max}}(M)\cong I^{\bar{p}}H_r(A)^*\) ( \(r\in \mathbb {N}\) ). If M is oriented and \(\bar{p}\ge \bar{n}\) , we also get \(H^r_{\mathrm{min}}(M)\cong I^{\bar{p}}H_r(A)\) . Thus our version of the Morse inequalities can be described in terms of \(I^{\bar{p}}H_*(A)\) . PubDate: 2018-01-13 DOI: 10.1007/s10455-017-9592-y

Authors:Tom Needham Abstract: Abstract We consider the space \({\mathcal {M}}\) of Euclidean similarity classes of framed loops in \({\mathbb {R}}^3\) . Framed loop space is shown to be an infinite-dimensional Kähler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of \({\mathcal {M}}\) by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of \(\mathcal {M}\) explicitly. Using this description, we show that \({\mathcal {M}}\) and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in \({\mathcal {M}}\) as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on \({\mathcal {M}}\) is shown to be Hamiltonian, and this is used to characterize the critical points of the weighted total twist functional. PubDate: 2018-01-13 DOI: 10.1007/s10455-018-9595-3

Authors:Almir Silva Santos; Maria Andrade Abstract: Abstract Our main goal in this work is to deal with results concern to the \(\sigma _2\) -curvature. First we find a symmetric 2-tensor canonically associated to the \(\sigma _2\) -curvature and we present an almost-Schur-type lemma. Using this tensor, we introduce the notion of \(\sigma _2\) -singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and \(\sigma _2\) -curvature. With a suitable condition on the \(\sigma _2\) -curvature, we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the three-dimensional torus does not admit a metric with constant scalar curvature and nonnegative \(\sigma _2\) -curvature unless it is flat. PubDate: 2018-01-10 DOI: 10.1007/s10455-018-9593-5

Authors:Daniele Angella; Hisashi Kasuya Pages: 363 - 411 Abstract: Abstract We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott–Chern cohomology. We are especially aimed at studying the Bott–Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precisely, we can construct explicit finite-dimensional double complexes that allow to compute the Bott–Chern cohomology of compact quotients of complex Lie groups, respectively, of some Lie groups of the type \(\mathbb {C}^n\ltimes _\varphi N\) where N is nilpotent. As an application, we compute the Bott–Chern cohomology of the complex parallelizable Nakamura manifold and of the completely solvable Nakamura manifold. In particular, the latter shows that the property of satisfying the \(\partial \overline{\partial }\) -Lemma is not strongly closed under deformations of the complex structure. PubDate: 2017-12-01 DOI: 10.1007/s10455-017-9560-6 Issue No:Vol. 52, No. 4 (2017)

Authors:Frederico Girão; Neilha M. Pinheiro Pages: 413 - 424 Abstract: Abstract We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov–Fenchel-type inequality for strictly convex hypersurfaces in the n-dimensional sphere, \(n \ge 3\) . PubDate: 2017-12-01 DOI: 10.1007/s10455-017-9562-4 Issue No:Vol. 52, No. 4 (2017)

Authors:Anna Siffert Pages: 425 - 456 Abstract: Abstract In this paper we show that the long-standing problem of classifying all isoparametric hypersurfaces in spheres with six different principal curvatures is still not complete. Moreover, we develop a structural approach that may be helpful for such a classification. Instead of working with the isoparametric hypersurface family in the sphere, we consider the associated Lagrangian submanifold of the real Grassmannian of oriented 2-planes in \({\mathbb {R}}^{n+2}\) . We obtain new geometric insights into classical invariants and identities in terms of the geometry of the Lagrangian submanifold. PubDate: 2017-12-01 DOI: 10.1007/s10455-017-9563-3 Issue No:Vol. 52, No. 4 (2017)

Authors:Martin de Borbon Pages: 457 - 464 Abstract: Abstract Let D be a smooth divisor in a compact complex manifold X and let \(\beta \in (0,1)\) . We use the liner theory developed by Donaldson (Essays in Mathematics and Its Applications, Springer, Berlin, pp 49–79, 2012) to show that in any positive co-homology class on X there is a Kähler metric with cone angle \(2\pi \beta \) along D which has bounded Ricci curvature. We use this result together with the Aubin–Yau continuity method to give an alternative proof of a well-known existence theorem for Kähler–Einstein metrics with cone singularities. PubDate: 2017-12-01 DOI: 10.1007/s10455-017-9565-1 Issue No:Vol. 52, No. 4 (2017)

Authors:Zejun Hu; Jiabin Yin; Zhenqi Li Abstract: Abstract The equivariant CR minimal immersions from the round 3-sphere \(S^3\) into the complex projective space \(\mathbb CP^n\) have been classified by the third author explicitly (Li in J Lond Math Soc 68:223–240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant and that all geometric properties of \(S^3=\mathrm{SU}(2)\) endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra \(\mathfrak {su}(2)\) , we establish an extended classification theorem for equivariant CR minimal immersions from the 3-sphere \(S^3\) into \(\mathbb CP^n\) without the assumption of constant sectional curvatures. PubDate: 2017-12-30 DOI: 10.1007/s10455-017-9590-0

Authors:Hui Ma; Andrey E. Mironov; Dafeng Zuo Abstract: 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: 2017-12-19 DOI: 10.1007/s10455-017-9589-6

Authors:Hilário Alencar; Adina Rocha Abstract: 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: 2017-12-13 DOI: 10.1007/s10455-017-9588-7

Authors:Oscar Macia; Yasuyuki Nagatomo Abstract: 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: 2017-12-11 DOI: 10.1007/s10455-017-9585-x

Authors:Rafael López Abstract: Abstract We classify all singular minimal surfaces in Euclidean space that are invariant by a uniparametric group of translations and rotations. PubDate: 2017-12-07 DOI: 10.1007/s10455-017-9586-9

Authors:Ignacio Bajo; Esperanza Sanmartín Abstract: 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: 2017-12-07 DOI: 10.1007/s10455-017-9587-8

Authors:Daniel Grady; Hisham Sati 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: 2017-11-24 DOI: 10.1007/s10455-017-9583-z

Authors:Diego Conti; Federico A. Rossi Abstract: 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: 2017-11-24 DOI: 10.1007/s10455-017-9584-y

Authors:Zhenxiao Xie; Tongzhu Li; Xiang Ma; Changping Wang Abstract: 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: 2017-11-20 DOI: 10.1007/s10455-017-9581-1

Authors:Daniel J. F. Fox Abstract: 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: 2017-11-20 DOI: 10.1007/s10455-017-9582-0

Authors:Liviu Ornea; Vladimir Slesar Abstract: 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: 2017-11-17 DOI: 10.1007/s10455-017-9579-8

Authors:Ivan Minchev; Jan Slovák Abstract: 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: 2017-11-05 DOI: 10.1007/s10455-017-9580-2

Authors:Pengshuai Shi Abstract: Abstract We compute the index of a Callias-type operator with APS boundary condition on a manifold with compact boundary in terms of combination of indexes of induced operators on a compact hypersurface. Our result generalizes the classical Callias-type index theorem to manifolds with compact boundary. PubDate: 2017-09-06 DOI: 10.1007/s10455-017-9575-z