43 research outputs found
A general comparison theorem for -harmonic maps in homotopy class
We prove a general comparison result for homotopic finite -energy
-harmonic maps between Riemannian manifolds, assuming that
is -parabolic and is complete and non-positively curved. In particular,
we construct a homotopy through constant -energy maps, which turn out to be
-harmonic when is compact. Moreover, we obtain uniqueness in the case of
negatively curved . This generalizes a well known result in the harmonic
setting due to R. Schoen and S.T. Yau.Comment: 19 page
Scalar curvature via local extent
We give a metric characterization of the scalar curvature of a smooth
Riemannian manifold, analyzing the maximal distance between points in
infinitesimally small neighborhoods of a point. Since this characterization is
purely in terms of the distance function, it could be used to approach the
problem of defining the scalar curvature on a non-smooth metric space. In the
second part we will discuss this issue, focusing in particular on Alexandrov
spaces and surfaces with bounded integral curvature.Comment: 22 pages. A new rigidity result has been added (see Proposition 17).
Some typos have been correcte
Stokes' theorem, volume growth and parabolicity
We present some new Stokes' type theorems on complete non-compact manifolds
that extend, in different directions, previous work by Gaffney and Karp and
also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity.
Applications to comparison and uniqueness results involving the p-Laplacian are
deduced.Comment: 15 pages. Corrected typos. Accepted for publication in Tohoku
Mathematical Journa
Lorentzian area measures and the Christoffel problem
We introduce a particular class of unbounded closed convex sets of
, called F-convex sets (F stands for future). To define them, we use
the Minkowski bilinear form of signature instead of the usual
scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic
space \H^d. Important examples are embeddings of the universal cover of
so-called globally hyperbolic maximal flat Lorentzian manifolds.
Basic tools are first derived, similarly to the classical study of convex
bodies. For example, F-convex sets are determined by their support function,
which is defined on \H^d. Then the area measures of order , are defined. As in the convex bodies case, they are the coefficients of the
polynomial in which is the volume of an approximation of
the convex set. Here the area measures are defined with respect to the
Lorentzian structure.
Then we focus on the area measure of order one. Finding necessary and
sufficient conditions for a measure (here on \H^d) to be the first area
measure of a F-convex set is the Christoffel Problem. We derive many results
about this problem. If we restrict to "Fuchsian" F-convex set (those who are
invariant under linear isometries acting cocompactly on \H^d), then the
problem is totally solved, analogously to the case of convex bodies. In this
case the measure can be given on a compact hyperbolic manifold.
Particular attention is given on the smooth and polyhedral cases. In those
cases, the Christoffel problem is equivalent to prescribing the mean radius of
curvature and the edge lengths respectively
Remarks on -vanishing results in geometric analysis
We survey some -vanishing results for solutions of Bochner or Simons
type equations with refined Kato inequalities, under spectral assumptions on
the relevant Schr\"{o}dinger operators. New aspects are included in the
picture. In particular, an abstract version of a structure theorem for stable
minimal hypersurfaces of finite total curvature is observed. Further geometric
applications are discussed.Comment: 18 pages. Some oversights corrected. Accepted for publication in
International Journal of Mathematic
Sobolev functions without compactly supported approximations
A basilar property and a useful tool in the theory of Sobolev spaces is the
density of smooth compactly supported functions in the space
(i.e. the functions with weak derivatives of orders to in ). On
Riemannian manifolds, it is well known that the same property remains valid
under suitable geometric assumptions. However, on a complete non-compact
manifold it can fail to be true in general, as we prove in this paper. This
settles an open problem raised for instance by E. Hebey [\textit{Nonlinear
analysis on manifolds: Sobolev spaces and inequalities}, Courant Lecture Notes
in Mathematics, vol. 5, 1999, pp. 48-49].Comment: 9 pages. Minor typos correcte
Density problems for second order Sobolev spaces and cut-off functions on manifolds with unbounded geometry
We consider complete non-compact manifolds with either a sub-quadratic growth
of the norm of the Riemann curvature, or a sub-quadratic growth of both the
norm of the Ricci curvature and the squared inverse of the injectivity radius.
We show the existence on such a manifold of a distance-like function with
bounded gradient and mild growth of the Hessian. As a main application, we
prove that smooth compactly supported functions are dense in . The
result is improved for avoiding both the upper bound on the Ricci tensor,
and the injectivity radius assumption. As further applications we prove new
disturbed Sobolev and Calder\'on-Zygmund inequalities on manifolds with
possibly unbounded curvature and highlight consequences about the validity of
the full Omori-Yau maximum principle for the Hessian.Comment: Improved version. As a main modification, we added a final Section 8
including some additional geometric applications of our result. Furthermore,
we proved in Section 7 a disturbed L^p-Sobolev-type inequality with weight
more general than the previous one. 25 pages. Comments are welcom