The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

Rigidity of Noncompact Finsler Manifolds

For a Euclidean space or a Minkowski space, we change the metric in a compact subset and show that the resulting Finsler manifold is isometric to the original standard space under certain conditions. We assume that the mean tangent curvature vanishes and the metric satisfies some curvature conditions or have no conjugate points.     

Poisson Equation on Some Complete Noncompact Manifolds

We study the Poisson equation on some complete noncompact manifolds with asymptotically nonnegative curvature. We will also study the limiting behavior of the nonhomogeneous heat equation on some complete noncompact manifolds with nonnegative curvature.     

Non-existence for Harmonic Maps on Complete Noncompact Manifolds

In this paper, we prove a nonexistence theorem on harmonic maps. This generalizes the well-known Liouville-type theorem on harmonic maps due to S.Y. Cheng and H.I Choi.     

Actions of Noncompact Semisimple Groups on Lorentz Manifolds

The above title is the same, but with “semisimple” instead of “simple,” as that of a notice by Nadine Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half of the announced results before her premature death. Here, using a different, geometric approach, we generalize her results to the semisimple case, and give proofs of all her announced results. Received: May 2006, Revision: February 2007, Accepted: March 2007     

Yamabe方程在完备流形上的一个推广

This paper considers a semilinear elliptic equation on a n-dimensional complete noncompact Riemannian manifold,which is a generalization of the well known Yamabe equation.An existence result is proved.     

Harmonic Maps from Complete Noncompact Manifolds

ln this paper we prove some general existence theorems of harmonic maps from complete noncompact manifolds with tho positive lower bounds of spectrum into convex balls. We solve the Dirichlet problem in classical domains and some special complete noncompact manifolds for harmonic maps into convex balls. We also study the existence of harmonic maps from some special complete noncompact manifolds into complete manifolds with nonpositive sectional curvature which are not simply connected.     

Transitive Actions on Lorentz Manifolds with Noncompact Stabilizer

If a topological group G acts on a topological space X, then we say that the action is orbit nonproper provided that, for some xX, the orbit map ggx:GX is nonproper. We consider the problem of classifying the connected, simply connected real Lie groups G admitting a locally faithful, orbit nonproper, isometric action on a connected Lorentz manifold. In an earlier paper, we found three collections of groups such that G admits such an action iff G is in one of the three collections. In another paper, we effectively described the first collection. In this paper, we show that the second collection contains a small, effectively described collection of groups, and, aside from those, it is contained in the union of the first and third collections. Finally, in a third paper, we effectively describe the third collection, thus solving the stated problem.     

关于非紧流形上的Ricci流的一个注记

设（M^3,90）是非紧三维Riemann流形,其Ricci曲率非负,单射半径有正的下界,且当x→∞时数量曲率R（x）→0。则以（M^3,go）为初始值的Ricci流在M^3×[0,∞）上有长期解。这推广了马和朱最近的一个结果．在高维情形我们也有相应的结果,并且我们给Chau,Tam和Yu在Ktihler情形的类似定理一个新的证明。     

Yamabe Flow and Myers Type Theorem on Complete Manifolds

In this paper, we prove the following Myers type theorem: If (M ⁿ ,g), n≥3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc≥?Rg, where R>0 is the scalar curvature and ?>0 is a uniform constant, then M ⁿ must be compact.     

Index Theorem for Equivariant Dirac Operators on Noncompact Manifolds

Let D be a (generalized) Dirac operator on a noncompact complete Riemannian manifold M acted on by a compact Lie group G. Let v: M g = Lie G be an equivariant map, such that the corresponding vector field on M does not vanish outside of a compact subset. These data define an element of K-theory of the transversal cotangent bundle to M. Hence, by embedding of M into a compact manifold, one can define a topological index of the pair (D,v) as an element of the completed ring of characters of G. We define an analytic index of (D,v) as an index space of certain deformation of D and we prove that the analytic and topological indexes coincide. As a main step of the proof, we show that index is an invariant of a certain class of cobordisms, similar to the one considered by Ginzburg, Guillemin and Karshon. In particular, this means that the topological index of Atiyah is also invariant under this class of noncompact cobordisms. As an application, we extend the Atiyah–Segal–Singer equivariant index theorem to our noncompact setting. In particular, we obtain a new proof of this theorem for compact manifolds.     

Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

Let (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ ₀ with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem

$$ (P_{\epsilon })\qquad -\mathcal {L}_{g} u+\epsilon u=u^{\frac {N+2}{N-2}}\ \text { in }\ (M,g) . $$

We prove that for any k ∈ ?, there exists ε _k > 0 such that for all ε ∈ (0, ε _k ) the problem (P _?? ) has a symmetric solution u _ε , which looks like the superposition of k positive bubbles centered at the point ξ ₀ as ε → 0. In particular, ξ ₀ is a towering blow-up point.

     

Atomic Decomposition of Hardy Type Spaces on Certain Noncompact Manifolds

In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the Hardy type spaces X ^k (M), introduced in a previous paper of the authors, have an atomic characterization. An atom in X ^k (M) is an atom in the Hardy space H ¹(M) introduced by Carbonaro, Mauceri, and Meda, satisfying an ??infinite dimensional?? cancellation condition. As an application, we prove that the Riesz transforms of even order $\nabla^{2k} \mathcal{L}^{-k}$ map X ^k (M) into L ¹(T _2k M).     

Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of Sⁿ by looking at the conormal bundle of appropriate submanifolds of Sⁿ. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson for submanifolds in Rⁿ in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G₂ and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy. Mathematics Subject Classification (2000): 53-XX, 58-XX.     

Ricci Flow on a Class of Noncompact Warped Product Manifolds

We consider the Ricci flow on noncompact \(n+1\)-dimensional manifolds M with symmetries, corresponding to warped product manifolds \(\mathbb {R}\times T^n\) with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate \(|{{\mathrm{Rm}}}|(p,t) \le C/t\) for some \(C > 0\) and all \(p \in M, t \in (1,\infty )\) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as \(t \rightarrow \infty \)) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the (n-dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.     

The Harnack Estimate for the Yamabe Flow on CR Manifolds of Dimension 3

We deform the contact form by the amount of the Tanaka–Webstercurvature on a closed spherical CR three-manifold.We show that if a contact form evolves with free torsion from initial datawith positive Tanaka–Webster curvature, then a certainHarnack inequality for the Tanaka–Webster curvature holds.     

Generalized Solutions to Boundary-Value Problems for Quasilinear Elliptic Equations on Noncompact Riemannian Manifolds

In the present work we develop approximation approach to evaluation of solutions to boundary-value problems for quasilinear equations of the elliptic type on arbitrary noncompact Riemannian manifolds. Our technique essentially bases on an approach from the papers of E. A. Mazepa and S. A. Korol’kov connected with introduction of equivalency classes of functions and representations. On the other hand, it generalizes the method of building of generalized solution to the Dirichlet problem for linear elliptic Laplace–Beltrami and Schrödinger equations in bounded domains in ? ⁿ , which is described in details in the works of M. V. Keldysh and E. M. Landis.     

Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds

We derive a sharp, localized version of elliptic type gradientestimates for positive solutions (bounded or not) to the heatequation. These estimates are related to the Cheng–Yauestimate for the Laplace equation and Hamilton's estimate forbounded solutions to the heat equation on compact manifolds.As applications, we generalize Yau's celebrated Liouville theoremfor positive harmonic functions to positive ancient (includingeternal) solutions of the heat equation, under certain growthconditions. Surprisingly this Liouville theorem for the heatequation does not hold even in Rⁿ without such a condition.We also prove a sharpened long-time gradient estimate for thelog of the heat kernel on noncompact manifolds. 2000 MathematicsSubject Classification 35K05, 58J35.     

Solvability of the Dirichlet Problem for the Poisson Equation on Some Noncompact Riemannian Manifolds

The behavior of solutions of the Poisson equation on noncompact Riemannian manifolds of a special form is studied. Sharp conditions for the unique solvability of the Dirichlet problem on the reconstruction of solutions of the Poisson equation from continuous boundary data at infinity are found.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.