777 research outputs found
Ricci surfaces
A Ricci surface is a Riemannian 2-manifold whose Gaussian curvature
satisfies . Every minimal surface isometrically
embedded in is a Ricci surface of non-positive curvature. At the
end of the 19th century Ricci-Curbastro has proved that conversely, every point
of a Ricci surface has a neighborhood which embeds isometrically in
as a minimal surface, provided . We prove this result in
full generality by showing that Ricci surfaces can be locally isometrically
embedded either minimally in or maximally in ,
including near points of vanishing curvature. We then develop the theory of
closed Ricci surfaces, possibly with conical singularities, and construct
classes of examples in all genera .Comment: 27 pages; final version, to appear in Annali della Scuola Normale
Superiore di Pisa - Classe di Scienz
Twistor Forms on Riemannian Products
We study twistor forms on products of compact Riemannian manifolds and show
that they are defined by Killing forms on the factors. The main result of this
note is a necessary step in the classification of compact Riemannian manifolds
with non-generic holonomy carrying twistor forms.Comment: 5 page
Adiabatic limits of eta and zeta functions of elliptic operators
We extend the calculus of adiabatic pseudo-differential operators to study
the adiabatic limit behavior of the eta and zeta functions of a differential
operator , constructed from an elliptic family of operators indexed by
. We show that the regularized values and
are smooth functions of at , and we identify
their values at with the holonomy of the determinant bundle, respectively
with a residue trace. For invertible families of operators, the functions
and are shown to extend smoothly to
for all values of . After normalizing with a Gamma factor, the zeta
function satisfies in the adiabatic limit an identity reminiscent of the
Riemann zeta function, while the eta function converges to the volume of the
Bismut-Freed meromorphic family of connection 1-forms.Comment: 32 pages, final versio
Fibered cusp versus - index theory
We prove that the indices of fibered-cusp and -Dirac operators on a spin
manifold with fibered boundary coincide if the associated family of Dirac
operators on the fibers of the boundary is invertible. This answers a question
raised by Piazza. Under this invertibility assumption, our method yields an
index formula for the Dirac operator of horn-cone and of fibered horn metrics.Comment: 8 pages, to appear in Rendiconti Semin. Mat. Parm
Compact lcK manifolds with parallel vector fields
We show that for a compact locally conformally K\"ahler manifold
carrying a non-trivial parallel vector field is either Vaisman,
or globally conformally K\"ahler, determined in an explicit way by some compact
K\"ahler manifold of dimension .Comment: 10 page
- …