Noncompact Manifolds Research Articles

𝑉-harmonic morphisms between Riemannian manifolds

A V V -harmonic morphism u : M → N u:M\to N between Riemannian manifolds is a smooth map which pulls back germs of harmonic functions on N N to germs of V V -harmonic functions on M M , where V V is a smooth vector field on M M . In this paper, we give some characterizations and examples of V V -harmonic morphisms. In addition, a dilation estimate and a Liouville-type theorem of V V -harmonic morphisms from noncompact complete manifolds are also established. As applications, we obtain the Liouville-type theorems for V V -harmonic morphisms from complete manifolds of nonnegative Bakry-Émery Ricci curvature, especially complete steady or shrinking Ricci solitons, to manifolds of dimension at least three or compact Riemann surface of genus at least two.

On solvability of the boundary value problems for harmonic function on noncompact Riemannian manifolds

On solvability of the boundary value problems for harmonic function on noncompact Riemannian manifolds

Profile decomposition in Sobolev spaces of non-compact manifolds

For many known non-compact embeddings of two Banach spaces $$E\hookrightarrow F$$, every bounded sequence in E has a subsequence that takes form of a profile decomposition—a sum of clearly structured terms with asymptotically disjoint supports plus a remainder that vanishes in the norm of F. In this paper we construct a profile decomposition for arbitrary sequences in the Sobolev space $$H^{1,2}(M)$$ of a Riemannian manifold with bounded geometry, relative to the embedding of $$H^{1,2}(M)$$ into $$L^{2^*}(M)$$, generalizing the well-known profile decomposition of Struwe (Math Z 187:511–517, 1984, Proposition 2.1) to the case of general bounded sequence and a non-compact manifold.

Wick rotation of the time variables for two-point functions on analytic backgrounds

We set up a general framework for Calder\'on projectors (and their generalization to non-compact manifolds), associated with complex Laplacians e.g. obtained by Wick rotation of a Lorentzian metric. In the analytic case, we use this to show that the Laplacian's Green's functions have analytic continuations whose boundary values are two-point functions of analytic Hadamard states. The result does not require the metric to be stationary. As an aside, we describe how thermal states are obtained as a special case of this construction if the coefficients are time-independent.

A characterization related to Schrödinger equations on Riemannian manifolds

In this paper, we consider the following problem: [Formula: see text] where [Formula: see text] is a [Formula: see text]-dimensional ([Formula: see text], non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, [Formula: see text] is a real parameter, [Formula: see text] is a positive coercive potential, [Formula: see text] is a bounded function and [Formula: see text] is a suitable nonlinearity. By using variational methods, we prove a characterization result for existence of solutions for [Formula: see text].

Higher-order representation stability and ordered configuration spaces of manifolds

Using the language of twisted skew-commutative algebras, we define \emph{secondary representation stability}, a stability pattern in the {\it unstable} homology of spaces that are representation stable in the sense of Church, Ellenberg, and Farb. We show that the rational homology of configuration spaces of ordered points in noncompact manifolds satisfies secondary representation stability. While representation stability for the homology of configuration spaces involves stabilizing by introducing a point ``near infinity,'' secondary representation stability involves stabilizing by introducing a pair of orbiting points -- an operation that relates homology groups in different homological degrees. This result can be thought of as a representation-theoretic analogue of \emph{secondary homological stability} in the sense of Galatius, Kupers, and Randal-Williams. In the course of the proof we establish some additional results: we give a new characterization of the homology of the complex of injective words, and we give a new proof of integral representation stability for configuration spaces of noncompact manifolds, extending previous results to nonorientable manifolds.

On VT-harmonic maps

VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat VT-harmonic maps and their parabolic analogues with PDE tools. We establish a Jäger–Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for VT-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.

Gradient estimates for a weighted nonlinear equation on complete noncompact manifolds

Gradient estimates for a weighted nonlinear equation on complete noncompact manifolds

On mirror maps for manifolds of exceptional holonomy

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities.

Euler flows and singular geometric structures

Tichler proved (Tischler D. 1970 Topology 9 , 153–154. ( doi:10.1016/0040-9383(70)90037-6 )) that a manifold admitting a smooth non-vanishing and closed one-form fibres over a circle. More generally, a manifold admitting k -independent closed one-form fibres over a torus T k . In this article, we explain a version of this construction for manifolds with boundary using the techniques of b -calculus (Melrose R. 1993 The Atiyah Patodi Singer index theorem . Research Notes in Mathematics. Wellesley, MA: A. K. Peters; Guillemin V, Miranda E, Pires AR. 2014 Adv. Math. ( N. Y. ) 264 , 864–896. ( doi:10.1016/j.aim.2014.07.032 )). We explore new applications of this idea to fluid dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold's structure theorem and describe b -symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities (Miranda E, Oms C. 2018 Contact structures with singularities. https://arxiv.org/abs/1806.05638 ) and what we call b -Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance Etnyre J, Ghrist R. 2000 Trans. Am. Math. Soc. 352 , 5781–5794. ( doi:10.1090/S0002-9947-00-02651-9 )). These results provide a new technique to analyse the geometry of steady fluid flows on non-compact manifolds with cylindrical ends. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

On the Sharp Dimension Estimate of CR Holomorphic Functions in Sasakian Manifolds

Abstract This is the very 1st paper to focus on the CR analogue of Yau’s uniformization conjecture in a complete noncompact pseudohermitian $(2n+1)$-manifold of vanishing torsion (i.e., Sasakian manifold), which is an odd dimensional counterpart of Kähler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.

Heat conservation for generalized Dirac Laplacians on manifolds with boundary

We consider a notion of conservation for the heat semigroup associated with a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitable way and imposing uniform lower bounds on the zero-order piece (Weitzenbock potential) of the Dirac Laplacian, and on the endomorphism defining the mixed boundary condition, we show that the corresponding conservation principle holds. A key ingredient in the proof is a domination property for the heat semigroup which follows from an extension to this setting of a Feynman–Kac formula recently proved by the author de Lima (Pac J Math 292(1):177–201, 2018) in the context of differential forms. When applied to the Hodge Laplacian acting on differential forms satisfying absolute boundary conditions, this extends previous results by Vesentini (Ann Math Pura Appl 182:1–19, 2003) and Masamune (Atti Accad Naz Lincei Rend Lincei Mat Appl 18(4):351–358, 2007) in the boundaryless case. Along the way, we also prove a vanishing result for $$L^2$$ harmonic sections in the broader context of generalized (not necessarily Dirac) Laplacians. These results are further illustrated with applications to the Dirac Laplacian acting on spinors and to the Jacobi operator acting on sections of the normal bundle of a free boundary minimal immersion.

Existence of minimal hypersurfaces in complete manifolds of finite volume

We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent interest: if a region U can be swept out by a family of hypersurfaces of volume at most V, then it can be swept out by a family of mutually disjoint hypersurfaces of volume at most $$V + \varepsilon $$.

Instantaneously complete Chern–Ricci flow and Kähler–Einstein metrics

In this work, we obtain some existence results of Chern–Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as $$t\rightarrow 0$$ . These results can be viewed as a generalization of an existence result of Ricci flow by Giesen and Topping for surfaces of hyperbolic type to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of Kahler-Einstein metric on complete non-compact Hermitian manifolds, which generalizes the work of Lott–Zhang and Tosatti–Weinkove to complete non-compact Hermitian manifolds with possibly unbounded curvature.

Homotopy classes of proper maps out of vector bundles

In this paper, we classify the homotopy classes of proper maps Erightarrow mathbb {R}^k, where E is a vector bundle over a compact Hausdorff space. As a corollary, we compute the homotopy classes of proper maps mathbb {R}^n rightarrow mathbb {R}^k. We find a stability range of such maps. We conclude with some remarks on framed submanifolds of non-compact manifolds, the relationship with proper homotopy classes of maps, and the Pontryagin–Thom construction.

Corrigendum to “Minimal surfaces in finite volume noncompact hyperbolic 3-manifolds”

Corrigendum to “Minimal surfaces in finite volume noncompact hyperbolic 3-manifolds”

The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth

Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of non-compact harmonic manifolds except for the flat spaces. Denote by $$h > 0$$ the mean curvature of horospheres in X, and set $$\rho = h/2$$ . Fixing a basepoint $$o \in X$$ , for $$\xi \in \partial X$$ , denote by $$B_{\xi }$$ the Busemann function at $$\xi $$ such that $$B_{\xi }(o) = 0$$ . Then for $$\lambda \in \mathbb {C}$$ the function $$e^{(i\lambda - \rho )B_{\xi }}$$ is an eigenfunction of the Laplace–Beltrami operator with eigenvalue $$-(\lambda ^2 + \rho ^2)$$ . For a function f on X, we define the Fourier transform of f by $$\begin{aligned} \tilde{f}(\lambda , \xi ) := \int _X f(x) e^{(-i\lambda -\rho )B_{\xi }(x)} \mathrm{{d}}vol(x) \end{aligned}$$ for all $$\lambda \in \mathbb {C}, \xi \in \partial X$$ for which the integral converges. We prove a Fourier inversion formula $$\begin{aligned} f(x) = C_0 \int _{0}^{\infty } \int _{\partial X} \tilde{f}(\lambda , \xi ) e^{(i\lambda - \rho )B_{\xi }(x)} \mathrm{{d}}\lambda _o(\xi ) |c(\lambda )|^{-2} \mathrm{{d}}\lambda \end{aligned}$$ for $$f \in C^{\infty }_c(X)$$ , where c is a certain function on $$\mathbb {R} - \{0\}$$ , $$\lambda _o$$ is the visibility measure on $$\partial X$$ with respect to the basepoint $$o \in X$$ and $$C_0 > 0$$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze–Stein phenomenon.

Yamabe Flow on Non-compact Manifolds with Unbounded Initial Curvature

We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $m\geq3$ under the assumption that the initial metric $g_0=u_0g_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor $u_0$ bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from above and below without growth condition.

Sharp gradient estimates for a heat equation in Riemannian manifolds

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation u t = Δ u + a u log ⁡ u u_t=\Delta u+au\log u in complete noncompact Riemannian manifolds. As its application, we show that if u u is a positive solution of the equation u t = Δ u u_t=\Delta u and log ⁡ u \log u is of sublinear growth in both spatial and time directions, then u u must be constant. This gradient estimate is sharp since it is well known that u ( x , t ) = e x + t u(x,t)=e^{x+t} satisfying u t = Δ u u_t=\Delta u .

Convergence of Ricci Flow on a Class of Warped Product Metrics

We consider Ricci flow starting from warped product manifolds \(\left( {\mathbb {R}}\times N, k_0 + g_0^2 g_N\right) \), whose typical fibre \((N,g_N)\) is closed and Ricci flat. Here \(k_0\) is a Riemannian metric on \({\mathbb {R}}\) and \(g_0: {\mathbb {R}}\rightarrow {\mathbb {R}}\) is positive. Under a mild condition, we show that (i) if the initial metric is asymptotic to the Ricci flat metric \(k_0 + c^2 g_N\), where \(c > 0\), the solution of the Ricci flow converges smoothly uniformly to a Ricci flat metric as \(t \rightarrow \infty \), up to pullback by a family of diffeomorphisms, and (ii) if the initial manifold is asymptotic to the real line, then the solution converges uniformly (in Gromov Hausdorff distance) to the real line as \(t \rightarrow \infty \). In the course of the proof, we establish an averaging and a convergence result for the heat equation on noncompact manifolds with time-dependent metric, that might be of independent interest.