Polar foliations and isoparametric maps

A singular Riemannian foliation F{\mathcal{F}} on a complete Riemannian manifold M is called a polar foliation if, for each regular point p, there is an immersed submanifold Σ, called section, that passes through p and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of M into the orbits of a polar action, i.e., an isometric action with sections. In this article we prove that the leaves of F{\mathcal{F}} coincide with the level sets of a smooth map H: M → Σ, if M is simply connected. In particular, the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.     

A new Lichnerowicz–Obata estimate in the presence of a parallel p-form

 Let (M ⁿ ,g) be a compact Riemannian manifold with a smooth boundary. In this paper, we give a Lichnerowicz-Obata type lower bound for the first eigenvalue of the Laplacian of (M ⁿ ,g) when M has a parallel p-form (2 ≤p≤n/2). This result follows from a new Bochner-Reilly's formula. Moreover, we give a characterization of the equality case when (M ⁿ ,g) is simply connected. Received: 1 June 2001     

A sharpL q-Liouville theorem forp-harmonic functions

We studyL ^q-Liouville properties of nonnegativep-superharmonic and, respectively,p-subharmonic functions on a complete Riemannian manifoldM. In particular, we prove that everyp-harmonic functionu ∈L ^q (M) is constant ifq>p−1. Supported by the Academy of Finland, Project 6355.     

<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-boundedness of Riesz transforms and imaginary powers of the Laplacian on Riemannian manifolds

We prove that the linearized Riesz transforms and the imaginary powers of the Laplacian areH ¹-bounded on complete Riemannian manifolds satisfying the doubling property and the Poincaré inequality, whereH ¹ denotes the Hardy space onM.     

über eine Klasse Riemannscher Mannigfaltigkeiten mit raumartigen pkf-ZusammenhÄngen

One considers m-dimensional Riemannian manifoldsM with tangent spaces T_p(M), p M that are a direct sum of a spacelike m-2 planeR _p and a 2-planeH _p. It is supposed that onM there exists a connection whose space-like components are parallel conformal flat (pkf). These components are generated by a vector field X. Assuming that X belongs to a pair X,Y of reciprocal quasi-cocircular vector fields and that the Pfaffian of this pair is the 1-form associated with the connection, the following results are derived: 1. X and Y are of equal constant length (This is true for all Riemannian manifolds). 2. The immersion of the integral manifold ofH _p intoM is cylindrical and the normal connection is flat. 3. The immersion of any space-like submanifold intoM is cylindrical with respect to the sections inH _p and umbilical with respect to all spacelike sections. 4. If m 4, the integral manifoldP _p is flat.

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Herrn Prof. Dr. WERNER BURAU zum 70.Geburtstag Klaus Buchner und Radu Rosca     

On m-Accretivity of Perturbed Bochner Laplacian in L p Spaces on Riemannian Manifolds

We consider a differential expression ${H=\nabla^*\nabla+V}We consider a differential expression H=?^*?+V{H=\nabla^*\nabla+V}, where ?{\nabla} is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L ^p (E), where 1 < p <  +∞. We study the same problem for the operator Δ _M  + V in L ^p (M), where 1 < p < ∞, Δ _M is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M.     

Eigenvalue estimate for a weighted p-Laplacian on compact manifolds with boundary

Let (M ⁿ , g) be a compact Riemannian manifold with convex boundary, let dμ = e ^h(x) dV (x) be a weighted measure on M, and let Δ_μ,p be the corresponding weighted p-Laplacian on M. We obtain a lower bound for the first nonzero Neumann eigenvalue of Δ_μ,p .     

Best constants for higher-order Rellich inequalities in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(Ω)

We obtain a series improvement to higher-order L ^p -Rellich inequalities on a Riemannian manifold M. The improvement is shown to be sharp as each new term of the series is added.      

The L p spectrum of Riemannian products

We prove that the L ^p spectrum of a Riemannian product M ₁×M ₂ coincides with the set theoretic sum of the L ^p spectra of M ₁ and M ₂ . Received: 13 June 2007     

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

An affine characterization of the Veronese surface

IfM ² is a nondegenerate surface in a 4-dimensional Riemannian manifold , then there is a natural affine metricg defined onM ². It is shown that this affine metricg is conformal to the induced Riemannian metric onM ² if and only ifM ² is a minimal submanifold of in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ⁴ or hyperbolic spaceH ⁴ whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM ² is a nondegenerate surface in the standard 4-sphereS ⁴ whose affine metric is homothetic to the induced Riemannian metric, thenM ² is a Veronese surface.T. Cecil was supported by NSF Grant No. DMS-9101961.     

Harmonic maps from a Riemannian manifold with a pole into an Hadamard manifold with negative sectional curvatures

In this paper we show a nonexistence result for harmonic maps with a rotational nondegeneracy condition from a Riemannian manifoldM with polep ₀ to a negatively curved Hadamard manifold under the condition that the metric tensor ofM is bounded and that the sectional curvature ofM at a pointp is bounded from below by −c dist(p ₀,p)⁻² (c: a positive constant) as dist(p ₀,p)→∞. Partly supported by Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan     

Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds

Consider a Riemannian manifold M which is a Galois covering of a compact manifold, with nilpotent deck transformation group G. For the Laplace operator on M, we prove a precise estimate for the gradient of the heat kernel, and show that the Riesz transforms are bounded in L^p(M), 1 < p < . We also obtain estimates for discrete oscillations of the heat kernel, and boundedness of discrete Riesz transform operators, which are defined using the action of G on M.Mathematics Subject Classification (2000): 58J35, 35B65, 42B20in final form: 8 August 2003     

Least Area Planes in Gromov Hyperbolic 3-Spaces with Co-compact Metric

Let M be a closed irreducible Riemannian 3-manifold such that π₁(M) is word hyperbolic, and p: X→ M the universal covering. Suppose that X has the Riemannian metric induced form that on M via p. In this paper, we will show that any Jordan curve in the boundary ∂X of X spans a properly embedded least area plane in X.Mathematics Subject Classiffications (2000). Primary 57M50; secondary 53A10     

The number of exceptional orbits of a pseudofree circle action on S 5

Let M be a closed Riemannian manifold of dimension 5 which admits a Riemannian metric of nonnegative sectional curvature. The aim of this short paper is to show that under certain lower bound of the orders of isotropy subgroups, every pseudofree and isometric S ¹-action on M cannot have more than five exceptional circle orbits. As a consequence, we conclude that a pseudofree and isometric S ¹-action on a 5-sphere S ⁵ with a Riemannian metric of nonnegative sectional curvature cannot have more than five exceptional circle orbits. This gives a result related to the Montgomery–Yang problem. In addition, we also give some further related result about nonnegatively curved manifolds of dimension 5 with an isometric but not necessarily pseudofree circle action.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

Geodesic loops and periodic geodesics on a Riemannian manifold diffeomorphic to <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

Let M ⁿ be a closed 2-connected Riemannian manifold, such that π₃(M ⁿ ) ≠ { 0 }. In this paper we prove that either there exists a periodic geodesic on M ⁿ of length ≤ 6d, where d is the diameter of M ⁿ , or at each point p ∈ M ⁿ there exists a geodesic loop of length ≤ 2d.     

Wavelet bases on a manifold

Using the decomposition method, we present in this paper constructions of multiresolution analyses on a compact Riemannian manifold M of dimension n(n∈N). These analyses are generated by a finite number of basic functions and are adapted to the study of the Sobolev spaces H¹(M) and .     

Blow-up examples for the Yamabe problem

It has been conjectured that if solutions to the Yamabe PDE on a smooth Riemannian manifold (M ⁿ , g) blow-up at a point p ? M{p \in M} , then all derivatives of the Weyl tensor W _g of g, of order less than or equal to [\fracn-62]{[\frac{n-6}{2}]} , vanish at p ? M{p \in M} . In this paper, we will construct smooth counterexamples to the Weyl Vanishing Conjecture for any n ≥ 25.