Deforming convex bodies in Minkowski geometry

We introduce and study certain deformation of Minkowski norms in [Formula: see text] determined by a set of [Formula: see text] linearly independent 1-forms and a smooth positive function of [Formula: see text] variables. In particular, the deformation of a Euclidean norm [Formula: see text] produces a Minkowski norm defined in our recent work; its indicatrix is a rotation hypersurface with a [Formula: see text]-dimensional axis passing through the origin. For [Formula: see text], our deformation generalizes the construction of [Formula: see text]-norms which form a rich class of “computable” Minkowski norms and play an important role in Finsler geometry. We characterize such pairs of a Minkowski norm and its image that Cartan torsions of the two norms either coincide or differ by a [Formula: see text]-reducible term. We conjecture that for [Formula: see text] any Minkowski norm can be approximated by images of a Euclidean norm.

A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Let Sm(c) be a Euclidean sphere of curvature c>0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).

On the principal Ricci curvatures of a Riemannian 3-manifold

We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (−μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.

Non-degeneracy of least-energy solutions for an elliptic problem with nearly critical nonlinearity

We consider the problem −Δu = c0K(x)upε, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth, bounded domain in ℝN, N ≥ 3, c0 = N(N − 2), pε = (N + 2)/(N − 2) − ε and K is a smooth, positive function on . We prove that least-energy solutions of the above problem are non-degenerate for small ε > 0 under some assumptions on the coefficient function K. This is a generalization of the recent result by Grossi for K ≡ 1, and needs precise estimates and a new argument.

The determination of caloric morphisms on Euclidean domains

LetDbe a domain in ℝm+1andEbe a domain in ℝn+1. A pair of a smooth mappingf:D → Eand a smooth positive function ϕ onDis called a caloric morphism if ϕ˙uofis a solution of the heat equation inDwheneveruis a solution of the heat equation inE. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case ofm < n, there are no caloric morphisms. In the case ofm = n, caloric morphisms are generated by the dilation, the rotation, the translation and the Appell transformation. In the case ofm > n, under some assumption onf, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ℝn+1.

SPECTRAL STUDY OF A COUPLED COMPACT-NONCOMPACT PROBLEM IN A NON-SELF-ADJOINT CASE

We consider a coupled problem of acoustic vibration of air in a porous medium Ω p that is in contact, by a plane interface Γ (x1=0), with free air in some region Ω f . The porous medium is made of infinitely narrow thin channels parallel to the x1-axis. In Refs. 1–3 and 5, problems of this type were considered with homogeneous Neumann boundary conditions on the exterior boundary of the whole domain Ω, Ω= Ω f ∪Γ∪Ω p . In this paper, we study the problem in the mentioned configuration but with the impedance condition ∂u/∂n=u/Z (where Z is a given complex number) on a part of the boundary of the porous medium Ω p defined by x1=–ℓ(x2, x3), where ℓ is a smooth positive function. Let us denote by A the operator associated with the coupled eigenvalue problem –Au=ω2u, and by Ap(x2, x3) the corresponding problem in a channel x2= const , x3= const . As in Ref. 1, two cases appear according to the values of ω2. In the first case ω2 is not an eigenvalue of Ap(x2, x3), and then we show that ω2 is either a point of the resolvent set or an isolated eigenvalue with finite multiplicity of A. In the second case ω2 is an eigenvalue of Ap(a2, a3) for some value (a2, a3) of (x2, x3), and then ω2 is a point of the essential spectrum of A. The novelty of this paper is the study of the spectrum of the operator A which is non self-adjoint and has a noncompact resolvent. We find a complex essential spectrum of A and we study its topological structure.