Remarks on singular elliptic theory for complete Riemannian manifolds

Nagano proved that if the non-homothetic conformal transformation between complete Riemannian manifolds with parallel Ricci tensor is admitted, then the manifolds are irreducible and isometric to a sphere. From this result and other reasons, it is natural to ask for the problem that does there exist globally a non-homothetic conformal transformation between complete product Riemannian manifolds? In this talk, we introduce and consider about this question and related topics.

In this paper we will show that on any complete noncompact Riemannian manifold with a finite volume there exist uncountably many geodesic loops of arbitrarily small length.

We present a classification of complete locally irreducible Riemannian manifolds with nonnegative curvature operator, which admit a nonzero and nondecomposable harmonic form with its square-integrable norm. We prove a vanishing theorem for harmonic forms on complete generic Riemannian manifolds with nonnegative curvature operator. We obtain similar results for closed and co-closed conformal Killing forms.

In this note we study a non-existence result of bi-harmonic maps from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $$\phi {:}\,(M,g)\rightarrow (N, h)$$ is a bi-harmonic map, where $$(M, g)$$ is a complete Riemannian manifold and $$(N,h)$$ a Riemannian manifold with non-positive sectional curvature. We prove: We also prove Liouville-type theorems on complete Riemannian manifolds and give applications of our non-existence result to bi-harmonic submersions.

Lefschetz Distribution of Lie Foliations.- Torsion, as a Function on the Space of Representations.- The K-theory of Twisted Group Algebras.- Twisted Burnside Theorem for Two-step Torsion-free Nilpotent Groups.- Ihara Zeta Functions for Periodic Simple Graphs.- Adiabatic Limits and the Spectrum of the Laplacian on Foliated Manifolds.- On the Non-standard Podle? Spheres.- Boundaries, Eta Invariant and the Determinant Bundle.- Elliptic Theory on Manifolds with Corners: I. Dual Manifolds and Pseudodifferential Operators.- Elliptic Theory on Manifolds with Corners: II. Homotopy Classification and K-Homology.- Dixmier Traceability for General Pseudo-differential Operators.- Topological Invariants of Bifurcation.- Modified Hochschild and Periodic Cyclic Homology.- L 2-invariants and Rank Metric.- Group Bundle Duality, Invariants for Certain C*-algebras, and Twisted Equivariant K-theory.- A New Topology on the Space of Unbounded Selfadjoint Operators, K-theory and Spectral Flow.

The purpose of this paper is to construct complete Riemannian Manifolds with negative Ricci curvature metric. We shall mostly concentrate on threedimensional manifolds. In [BJ], Bourguignon raised the problem that the connected sum of two Riemannian Manifolds with negative Ricci curvature has a metric with negative Ricci curvature. Yau first observed that the connected sum of two hyperbolic manifolds has a negative Ricei curvature metric. In the first part of the paper, we shall give the affirmative answer to the above question. We actually prove something more. We shall prove that the "connected sum along circles" of two Riemannian Manifolds with negative Ricci curvature metric (Sect. 1) has a negative Ricci curvature metric. In the second part, we only consider three-dimensional manifolds. We use the above results and some topological constructions to obtain new manifolds with negative Ricci curvature metric from a given manifold with negative Ricci curvature metric. Give M a complete Riemannian three-dimensional manifold with negative Ricci curvature metric, then there are complete negative Ricci curvature Riemannian metric on M # S 2 x S I ; M ~ S 2 x S I # L ( p , q ) , and M ~ S 2 x S ~ 1; x S 1, where 2; is any oriented Riemann surface. For the motivation of this paper see [YS]. This paper is an extended version of the second part of the author's Ph.D. thesis at Stony Brook. I would like to thank my advisor Professor Blaine Lawson for suggesting this problem to me, and inspiring discussions and valuable advice.

Properly immersed submanifolds in complete Riemannian manifolds

In this paper, we give a barrier argument at infinity for solutions of an elliptic equation on a complete Riemannian manifold. By using the barrier argument, we can construct a nonnegative (bounded, respectively) solution of the elliptic equation, which takes the given data at infinity of each end. In particular, we prove that if a complete Riemannian manifold has finitely many ends, each of which is Harnack and nonparabolic, then the set of bounded solutions of the elliptic equation is finite dimensional, in some sense. We also prove that if a complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincare inequality and the finite covering condition on each end, then there exists a nonnegative solution of an elliptic equation taking the given data at infinity of each end of the manifold. These results generalize those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Holopainen, and of the present authors, but with the barrier argument at infinity that enables one to overcome the obstacle due to the nonlinearity of solutions.

In this paper, we prove that the dimension of the space of positive (bounded, respectively) \(\mathcal{L}\)-harmonic functions on a complete Riemannian manifold with \(\mathcal{L}\)-regular ends is equal to the number of ends (\(\mathcal{L}\)-nonparabolic ends, respectively). This result is a solution of an open problem of Grigor'yan related to the Liouville property for the Schrodinger operator \(\mathcal{L}\). We also prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincare inequality and the finite covering condition on each end, then the dimension of the space of positive (bounded, respectively,) solutions for the Schrodinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends (\(\mathcal{L}\)-nonparabolic ends, respectively). This is a partial answer of the question, suggested by Li, related to the regularity of ends of a complete Riemannian manifold. Especially, our results directly generalize various earlier results of Yau, of Li and Tam, of Grigor'yan, and of present authors, but with different techniques that the peculiarity of the Schrodinger operator demands.

Hamilton’s Harnack inequality and the [formula omitted]-entropy formula on complete Riemannian manifolds

Examples of edge operators include Laplacians on asymptotically flat and asymptotically hyperbolic manifolds. Edge operators also arise in boundary problems around higher condimension boundaries. This paper is concerned with the analysis of general elliptic edge operators with constant indicide roots. We determine when such an operator has a distributional asymptotic expansion. Conditions are given to guarantee that the coefficients of this expansion are smooth. In Part I of this paper we only study the case when the operator is semi-Fredholm. Part II will examine edge operators with infinite dimensional kernel and cokernel, as well as develop the theory of Poisson edge operators.

In this short note we study a nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $${\phi : (M, g) \to (N, h)}$$ is a biharmonic map, where (M, g) is a complete Riemannian manifold and (N, h) a Riemannian manifold with nonpositive sectional curvature, we will prove that $${\phi}$$ is a harmonic map if one of the following conditions holds: (i) $${|d\phi|}$$ is bounded in L q (M) and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 \leq q \leq \infty}$$ , $${1 < p < \infty}$$ ; or (ii) $${Vol(M) = \infty}$$ and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 < p < \infty}$$ . In addition, if N has strictly negative sectional curvature, we assume that $${rank\phi(q) \geq 2}$$ for some $${q \in M}$$ and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 < p < \infty}$$ . These results improve the related theorems due to Baird et al. (cf. Ann Golb Anal Geom 34:403–414, 2008), Nakauchi et al. (cf. Geom. Dedicata 164:263–272, 2014), Maeta (cf. Ann Glob Anal Geom 46:75–85, 2014), and Luo (cf. J Geom Anal 25:2436–2449, 2015).

Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world “resolution” is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.

Consider the tri-harmonic differential expression $L_{V}^{\nabla}u=\left(\nabla^{+}\nabla\right)^{3}u+Vu$, on sections of a hermitian vector bundle over a complete Riemannian manifold $\left(M,g\right)$ with metric $g$, where $\nabla$ is a metric covariant derivative on bundle E over complete Riemannian manifold, $\nabla^{+}$ is the formal adjoint of $\nabla$ and $V$ is a self adjoint bundle on $E$. We will give conditions for $L_{V}^{\nabla}$ to be essential self-adjoint in $L^{2}\left(E\right).$ Additionally, we provide sufficient conditions for $L_{V}^{\nabla}$ to be separated in $L^{2}\left( E\right)$. According to Everitt and Giertz, the differential operator $L_{V}^{\nabla}$ is said to be separated in $L^{2}\left( E\right) $ if for all $u$ $\in L^{2}\left( E\right)$ such that $L_{V}^{\nabla}u\in L^{2}\left( E\right) $, we have $Vu\in L^{2}\left( E\right)$.

By a classical theorem of Gallot (Ann. Sci. Éc. Norm. Sup. 12: 235–267, 1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-Kählerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.