Authors:A. Barros; A. Da Silva Pages: 1 - 9 Abstract: Abstract The aim of this note is to present some results about critical metrics for quadratic functional \({\mathcal {F}}_{t,s}\) defined on \({\mathcal {M}}_{1}=\{g\in {\mathcal {M}} \mathrm{Vol}(g)=1\}\) , where \({\mathcal {M}}\) is the space of smooth Riemannian metrics on a closed smooth Riemannian manifold \(M^n\) . We know that space form metrics are critical point for \({\mathcal {F}}_{t,s}\) . We investigate when the converse is true. In particular, we show that locally conformally flat critical metrics with some additional conditions are space form metrics. PubDate: 2017-07-01 DOI: 10.1007/s10455-016-9541-1 Issue No:Vol. 52, No. 1 (2017)
Authors:Martins Bruveris Pages: 11 - 24 Abstract: Abstract Let \(F : H^q \rightarrow H^q\) be a \(C^k\) -map between Sobolev spaces, either on \({{\mathbb {R}}}^d\) or on a compact manifold. We show that equivariance of F under the diffeomorphism group allows to trade regularity of F as a nonlinear map for regularity in the image space: for \(0 \le l \le k\) , the map \(F: H^{q+l} \rightarrow H^{q+l}\) is well defined and of class \(C^{k-l}\) . This result is used to study the regularity of the geodesic boundary value problem for Sobolev metrics on the diffeomorphism group and the space of curves. PubDate: 2017-07-01 DOI: 10.1007/s10455-017-9544-6 Issue No:Vol. 52, No. 1 (2017)
Authors:Tian Chong; Yuxin Dong; Yibin Ren Pages: 25 - 44 Abstract: Abstract In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps. PubDate: 2017-07-01 DOI: 10.1007/s10455-017-9547-3 Issue No:Vol. 52, No. 1 (2017)
Authors:Burcu Bektaş; Joeri Van der Veken; Luc Vrancken Pages: 45 - 55 Abstract: Abstract We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations. PubDate: 2017-07-01 DOI: 10.1007/s10455-017-9548-2 Issue No:Vol. 52, No. 1 (2017)
Authors:Benjamin Küster Pages: 57 - 97 Abstract: Abstract We study the functional calculus for operators of the form \(f_h(P(h))\) within the theory of semiclassical pseudodifferential operators, where \(\{f_h\}_{h\in (0,1]}\subset \mathrm{C^\infty _c}({{\mathbb {R}}})\) denotes a family of h-dependent functions satisfying some regularity conditions, and P(h) is either an appropriate self-adjoint semiclassical pseudodifferential operator in \(\mathrm{L}^2({{\mathbb {R}}}^n)\) or a Schrödinger operator in \(\mathrm{L}^2(M), M\) being a closed Riemannian manifold of dimension n. The main result is an explicit semiclassical trace formula with remainder estimate that is well-suited for studying the spectrum of P(h) in spectral windows of width of order \(h^\delta \) , where \(0\le \delta <\frac{1}{2}\) . PubDate: 2017-07-01 DOI: 10.1007/s10455-017-9549-1 Issue No:Vol. 52, No. 1 (2017)
Authors:Dmitri Alekseevsky; Fabio Zuddas Pages: 99 - 128 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: 2017-07-01 DOI: 10.1007/s10455-017-9550-8 Issue No:Vol. 52, No. 1 (2017)
Authors:Ly Kim Ha Pages: 327 - 346 Abstract: 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: 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: 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: 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: 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: 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: 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:Daniele Angella; Alexandra Otiman; Nicoletta Tardini Abstract: Abstract We study the Morse–Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz condition. We consider solvmanifolds and Oeljeklaus–Toma manifolds. In particular, we prove that Oeljeklaus–Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type \(S^0\) . PubDate: 2017-07-17 DOI: 10.1007/s10455-017-9568-y
Authors:Frederico Girão; Neilha M. Pinheiro 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: 2017-07-14 DOI: 10.1007/s10455-017-9562-4
Authors:Bart Dioos; Luc Vrancken; Xianfeng Wang Abstract: Abstract In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) . We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\) . We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\) . PubDate: 2017-06-28 DOI: 10.1007/s10455-017-9567-z
Authors:Andreas Savas-Halilaj; Knut Smoczyk Abstract: Abstract In this article, we give a complete description of the evolution of an area decreasing map \(f{: }M\rightarrow N\) , induced by the mean curvature of their graph, in the situation where M and N are complete Riemann surfaces with bounded geometry, M being compact, for which their sectional curvatures \(\sigma _M\) and \(\sigma _N\) satisfy \(\min \sigma _M\ge \sup \sigma _N\) . PubDate: 2017-06-22 DOI: 10.1007/s10455-017-9566-0
Authors:Włodzimierz Jelonek Abstract: Abstract We give a description of all Gray AC manifolds (M, g) whose Ricci tensor has two eigenvalues of multiplicity 1 and \(\dim M-1\) . PubDate: 2017-06-21 DOI: 10.1007/s10455-017-9564-2
Authors:Martin de Borbon 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: 2017-06-17 DOI: 10.1007/s10455-017-9565-1
Authors:Anna Siffert 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: 2017-06-13 DOI: 10.1007/s10455-017-9563-3
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