Authors:Ly Kim Ha Pages: 327 - 346 Abstract: Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \) . Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \) . We study the Poincaré–Lelong equation $$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$ in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\) -equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \) . PubDate: 2017-06-01 DOI: 10.1007/s10455-016-9537-x Issue No:Vol. 51, No. 4 (2017)
Authors:Indranil Biswas; Arjun Paul Pages: 347 - 358 Abstract: Let X be a connected complex manifold equipped with a holomorphic action of a complex Lie group G. We investigate conditions under which a principal bundle on X admits a G-equivariance structure. PubDate: 2017-06-01 DOI: 10.1007/s10455-016-9538-9 Issue No:Vol. 51, No. 4 (2017)
Authors:Huihong Jiang; Yi-Hu Yang Pages: 359 - 366 Abstract: In this note, we show that a complete n-dim Riemannian manifold with nonnegative Ricci curvature is of finite topological type provided that the diameter growth of M is of order \(o(r^{((n-1)\alpha +1)/n})\) and the sectional curvature is no less than \(-{\frac{c}{r^{2\alpha }}}\) (here, \(0 \le \alpha \le 1\) and c is some positive constant) outside a geodesic ball large enough. In particular, if in a neighborhood of an isolated end of the manifold in question, the above assumptions are satisfied, then the end has a collared neighborhood. PubDate: 2017-06-01 DOI: 10.1007/s10455-016-9539-8 Issue No:Vol. 51, No. 4 (2017)
Authors:Mihai Băileşteanu Pages: 367 - 378 Abstract: We prove a differential Harnack inequality for the solution of the parabolic Allen–Cahn equation \( \frac{\partial f}{\partial t}=\triangle f-(f^3-f)\) on a closed n-dimensional manifold. As a corollary, we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation. PubDate: 2017-06-01 DOI: 10.1007/s10455-016-9540-2 Issue No:Vol. 51, No. 4 (2017)
Authors:Jinwoo Shin Pages: 379 - 399 Abstract: In this paper we study four-dimensional \((m,\rho )\) -quasi-Einstein manifolds with harmonic Weyl curvature when \(m\notin \{0,\pm 1,-2,\pm \infty \}\) and \(\rho \notin \{\frac{1}{4},\frac{1}{6}\}\) . We prove that a non-trivial \((m,\rho )\) -quasi-Einstein metric g (not necessarily complete) is locally isometric to one of the following: (i) \({\mathcal {B}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\) , where \({\mathcal {B}}^2_\frac{R}{2(m+2)}\) is the northern hemisphere in the two-dimensional (2D) sphere \({\mathbb {S}}^2_\frac{R}{2(m+2)}\) , \({\mathbb {N}}_\delta \) is a 2D Riemannian manifold with constant curvature \(\delta \) , and R is the constant scalar curvature of g. (ii) \({\mathcal {D}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\) , where \({\mathcal {D}}^2_\frac{R}{2(m+2)}\) is half (cut by a hyperbolic line) of hyperbolic plane \({\mathbb {H}}^2_\frac{R}{2(m+2)}\) . (iii) \({\mathbb {H}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\) . (iv) A certain singular metric with \(\rho =0\) . (v) A locally conformal flat metric. By applying this local classification, we obtain a classification of the complete \((m,\rho )\) -quasi-Einstein manifolds given the condition of a harmonic Weyl curvature. Our result can be viewed as a local classification of gradient Einstein-type manifolds. A corollary of our result is the classification of \((\lambda ,4+m)\) -Einstein manifolds, which can be viewed as (m, 0)-quasi-Einstein manifolds. PubDate: 2017-06-01 DOI: 10.1007/s10455-017-9542-8 Issue No:Vol. 51, No. 4 (2017)
Authors:Farid Madani; Andrei Moroianu; Mihaela Pilca Pages: 401 - 417 Abstract: We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is \(-\infty \) , and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold. PubDate: 2017-06-01 DOI: 10.1007/s10455-017-9545-5 Issue No:Vol. 51, No. 4 (2017)
Authors:B. Kruglikov; H. Winther Pages: 419 - 421 Abstract: We correct an error in the second part of Theorem 3 of our original paper. PubDate: 2017-06-01 DOI: 10.1007/s10455-017-9546-4 Issue No:Vol. 51, No. 4 (2017)
Authors:Wen Lu Pages: 231 - 244 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:Francesco Bei; Batu Güneysu Pages: 267 - 286 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: 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: 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:Martin de Borbon 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: 2017-06-17 DOI: 10.1007/s10455-017-9565-1
Authors:Anna Siffert 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: 2017-06-13 DOI: 10.1007/s10455-017-9563-3
Authors:B. S. Kruglikov; H. Winther Abstract: For an almost product structure J on a manifold M of dimension 6 with non-degenerate Nijenhuis tensor \(N_J\) , we show that the automorphism group \(G=\mathrm{Aut}(M,J)\) has dimension at most 14. In the case of equality G is the exceptional Lie group \(G_2^*\) . The next possible symmetry dimension is proved to be equal to 10, and G has Lie algebra \(\mathfrak {sp}(4,{\mathbb R})\) . Both maximal and submaximal symmetric structures are globally homogeneous and strictly nearly para-Kähler. We also demonstrate that whenever the symmetry dimension is at least 9, then the automorphism algebra acts locally transitively. PubDate: 2017-05-20 DOI: 10.1007/s10455-017-9561-5
Authors:Yuriĭ Gennadievich Nikonorov Abstract: The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterized by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic orbit Riemannian space, we describe the structure of the nilradical and the radical of the Lie algebra of the isometry group. In the final part, we discuss some new tools to study geodesic orbit Riemannian spaces, related to compact Lie group representations with non-trivial principal isotropy algebras. We discuss also some new examples of geodesic orbit Riemannian spaces, new methods to obtain such examples, and some unsolved questions. PubDate: 2017-05-06 DOI: 10.1007/s10455-017-9558-0
Authors:Hendrik Herrmann; Chin-Yu Hsiao; Xiaoshan Li Abstract: Let X be a compact strongly pseudoconvex CR manifold with a transversal CR \(S^1\) -action. In this paper, we establish the asymptotic expansion of Szegő kernels of positive Fourier components, and by using the asymptotics, we show that X can be equivariant CR embedded into some \(\mathbb {C}^N\) equipped with a simple \(S^1\) -action. An equivariant embedding of quasi-regular Sasakian manifold is also derived. PubDate: 2017-05-03 DOI: 10.1007/s10455-017-9559-z
Authors:Daniele Angella; Hisashi Kasuya 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: 2017-05-03 DOI: 10.1007/s10455-017-9560-6
Authors:D. Perrone 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: 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: 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
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