Authors:Wen Lu Pages: 231 - 244 Abstract: Abstract In this paper, we establish the canonical isomorphism between the Witten instanton complex and the Thom-Smale complex on manifolds with boundary with arbitrary Riemannian metric using Bismut-Lebeau’s analytic localization techniques. PubDate: 2017-04-01 DOI: 10.1007/s10455-016-9532-2 Issue No:Vol. 51, No. 3 (2017)

Authors:Anton S. Galaev Pages: 245 - 265 Abstract: Abstract It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-Kählerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group. PubDate: 2017-04-01 DOI: 10.1007/s10455-016-9533-1 Issue No:Vol. 51, No. 3 (2017)

Authors:Francesco Bei; Batu Güneysu Pages: 267 - 286 Abstract: Abstract The main result of this paper is a sufficient condition to have a compact Thom–Mather stratified pseudomanifold endowed with a \(\hat{c}\) -iterated edge metric on its regular part q-parabolic. Moreover, besides stratified pseudomanifolds, the q-parabolicity of other classes of singular spaces, such as compact complex Hermitian spaces, is investigated. PubDate: 2017-04-01 DOI: 10.1007/s10455-016-9534-0 Issue No:Vol. 51, No. 3 (2017)

Authors:Julien Roth; Julian Scheuer Pages: 287 - 304 Abstract: Abstract We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to r-stability as well as to almost-Einstein hypersurfaces. PubDate: 2017-04-01 DOI: 10.1007/s10455-016-9535-z Issue No:Vol. 51, No. 3 (2017)

Authors:Grant Cairns; Ana Hinić Galić; Yuri Nikolayevsky Pages: 305 - 325 Abstract: Abstract This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra \(\mathfrak {g}\) (respectively, of the Grassmannian of two-planes of \(\mathfrak {g}\) ) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on \(\mathfrak {g}\) . In the second part we study the subsets of \(\mathfrak {g}\) which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closure of these subsets is the whole algebra \(\mathfrak {g}\) , apart from two exceptional cases: when \(\mathfrak {g}\) is two-step nilpotent and when \(\mathfrak {g}\) contains a codimension one abelian ideal. PubDate: 2017-04-01 DOI: 10.1007/s10455-016-9536-y Issue No:Vol. 51, No. 3 (2017)

Authors:Scott Van Thuong Pages: 109 - 128 Abstract: Abstract We classify left invariant metrics on the 4-dimensional, simply connected, unimodular Lie groups up to automorphism. When the corresponding Lie algebra is of type (R), this is equivalent to classifying the left invariant metrics up to isometry, but in general the classification up to automorphism is finer than that up to isometry. In the abelian case, all left invariant metrics are isometric. In the nilpotent case, the space of metrics can have dimension 1 or 3. In the solvable case, the dimension can be 2, 4, or 5. There are two non-solvable 4-dimensional unimodular groups, and the space of metrics has dimension 6 in both of these cases. PubDate: 2017-03-01 DOI: 10.1007/s10455-016-9527-z Issue No:Vol. 51, No. 2 (2017)

Authors:R. Albuquerque Pages: 129 - 154 Abstract: Abstract We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle \(E\longrightarrow M\) , over a Riemannian manifold M, when E is endowed with a metric connection. The tangent bundle of E admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of E; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant–Salamon type \({\mathrm {G}_{2}}\) manifolds. PubDate: 2017-03-01 DOI: 10.1007/s10455-016-9528-y Issue No:Vol. 51, No. 2 (2017)

Authors:Hisashi Kasuya Pages: 155 - 177 Abstract: Abstract We describe the generalized Kuranishi spaces of solvmanifolds with left-invariant complex structures. By using such description, we study the stability of left-invariantness of deformed generalized complex structures and smoothness of generalized Kuranishi spaces on certain classes of solvmanifolds. We also give explicit finite-dimensional cochain complexes which computes the holomorphic Poisson cohomology of nilmanifolds and solvmanifolds. PubDate: 2017-03-01 DOI: 10.1007/s10455-016-9529-x Issue No:Vol. 51, No. 2 (2017)

Authors:Giuseppe Pipoli; Carlo Sinestrari Pages: 179 - 188 Abstract: Abstract We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere. PubDate: 2017-03-01 DOI: 10.1007/s10455-016-9530-4 Issue No:Vol. 51, No. 2 (2017)

Authors:Vanderson Lima Pages: 189 - 208 Abstract: Abstract Unlike \(\mathbb {R}^{3}\) , the homogeneous spaces \(\mathbb {E}(-1,\tau )\) have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantee that a minimal surface in \(\mathbb {E}(-1,\tau )\) is such a graph. More specifically, we introduce the definition of a generalized slab in \(\mathbb {E}(-1,\tau )\) and prove that a properly immersed minimal surface of finite topology inside such a slab region has multi-graph ends. Moreover, when the surface is embedded, the ends are graphs. When the surface is embedded and simply connected, it is an entire graph. PubDate: 2017-03-01 DOI: 10.1007/s10455-016-9531-3 Issue No:Vol. 51, No. 2 (2017)

Authors:D. Perrone Abstract: Abstract Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), \({\widetilde{SL}}(2,{\mathbb {R}})\) , \({\widetilde{E}}(2)\) , E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure \((\eta _1,\eta _2)\) with the so-called Webster function \({\mathcal {W}}\) constant along the flow of \(\xi _1\) (equivalently \(\xi _2\) ). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)). PubDate: 2017-04-08 DOI: 10.1007/s10455-017-9555-3

Authors:Israel Evangelista; Keomkyo Seo Abstract: Abstract In this paper, we estimate the p-fundamental tone of submanifolds in a Cartan–Hadamard manifold. First, we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension-one \(C^2\) -foliations of open subsets \(\Omega \) of Riemannian manifolds M and obtain lower bound estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of \(\Omega \) . PubDate: 2017-04-07 DOI: 10.1007/s10455-017-9557-1

Authors:Matthias Ludewig Abstract: Abstract We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of m-Laplace type operators L on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns out to be given by the local zeta function of L. In particular, the constant term in the asymptotic expansion of the Green’s function \(L^{-1}\) is often called the mass of L, which (in case that L is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant m-Laplace operators L (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of M is odd and that \(\ker L = 0\) , and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied. PubDate: 2017-04-07 DOI: 10.1007/s10455-017-9556-2

Authors:Vincent Bonini; Jie Qing; Jingyong Zhu Abstract: Abstract In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces \(\phi :M^n \rightarrow \mathbb {H}^{n+1}\) and a class of conformal metrics on domains of the round sphere \(\mathbb {S}^n\) . Some of the key aspects of the correspondence and its consequences have dimensional restrictions \(n\ge 3\) due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of \(\mathbb {S}^n\) . In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of Bonini et al. (2015) to all dimensions \(n\ge 2\) in a unified way. In the case of a single point boundary \(\partial _{\infty }\phi (M)=\{x\} \subset \mathbb {S}^n\) , we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in \(\mathbb {H}^{3}\) . PubDate: 2017-03-30 DOI: 10.1007/s10455-017-9554-4

Authors:Dorothee Schueth Abstract: Abstract If G is a compact Lie group endowed with a left invariant metric g, then G acts via pullback by isometries on each eigenspace of the associated Laplace operator \(\Delta _g\) . We establish algebraic criteria for the existence of left invariant metrics g on G such that each eigenspace of \(\Delta _g\) , regarded as the real vector space of the corresponding real eigenfunctions, is irreducible under the action of G. We prove that generic left invariant metrics on the Lie groups \(G={ SU}(2)\times \cdots \times { SU}(2)\times T\) , where T is a (possibly trivial) torus, have the property just described. The same holds for quotients of such groups G by discrete central subgroups. In particular, it also holds for \({ SO}(3)\) , \({ U}(2)\) , \({ SO}(4)\) . PubDate: 2017-03-30 DOI: 10.1007/s10455-017-9553-5

Authors:Lana Casselmann Abstract: Abstract We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free \(S^1\) -action, the \(S^1\) -quotient is a symplectic manifold, and our result reproduces Kirwan’s surjectivity for these symplectic manifolds. We further prove a Tolman–Weitsman type description of the kernel of the basic Kirwan map for \(S^1\) -actions and show that torus actions on a K-contact manifold that preserve the contact form and admit 0 as a regular value of the contact moment map are equivariantly formal in the basic setting. PubDate: 2017-03-22 DOI: 10.1007/s10455-017-9552-6

Authors:Wanjun Ai; Hao Yin Abstract: Abstract We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy. PubDate: 2017-03-20 DOI: 10.1007/s10455-017-9551-7

Authors:Dmitri Alekseevsky; Fabio Zuddas Abstract: Abstract Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\) . Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\) , where \(K = N_G(L)\) , endowed with an invariant complex structure \(J^F\) . We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics. PubDate: 2017-03-11 DOI: 10.1007/s10455-017-9550-8

Authors:Gabriela Tereszkiewicz; Maciej P. Wojtkowski Abstract: Abstract We study homogeneous Weyl connections with non-positive sectional curvatures. The Cartesian product \({\mathbb S}^1 \times M\) carries canonical families of Weyl connections with such a property, for any compact Riemmanian manifold M. We prove that if a homogeneous Weyl connection on a manifold, modeled on a unimodular Lie group, is non-positive in a stronger sense (stretched non-positive), then it must be locally of the product type. PubDate: 2017-02-27 DOI: 10.1007/s10455-016-9526-0

Authors:B. Kruglikov; H. Winther Abstract: Abstract We correct an error in the second part of Theorem 3 of our original paper. PubDate: 2017-02-23 DOI: 10.1007/s10455-017-9546-4