30 research outputs found
A negative mass theorem for surfaces of positive genus
We define the "sum of squares of the wavelengths" of a Riemannian surface
(M,g) to be the regularized trace of the inverse of the Laplacian. We normalize
by scaling and adding a constant, to obtain a "mass", which is scale invariant
and vanishes at the round sphere. This is an anlaog for closed surfaces of the
ADM mass from general relativity. We show that if M has positive genus then on
each conformal class, the mass attains a negative minimum. For the minimizing
metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a
Moser-Trudinger-Onofri type inequality.Comment: 8 page
Multiplicative anomaly and zeta factorization
Some aspects of the multiplicative anomaly of zeta determinants are
investigated. A rather simple approach is adopted and, in particular, the
question of zeta function factorization, together with its possible relation
with the multiplicative anomaly issue is discussed. We look primordially into
the zeta functions instead of the determinants themselves, as was done in
previous work. That provides a supplementary view, regarding the appearance of
the multiplicative anomaly. Finally, we briefly discuss determinants of zeta
functions that are not in the pseudodifferential operator framework.Comment: 20 pages, AIP styl
Lower order terms in Szego type limit theorems on Zoll manifolds
This is a detailed version of the paper math.FA/0212273. The main motivation
for this work was to find an explicit formula for a "Szego-regularized"
determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll
manifold. The idea of the Szego-regularization was suggested by V. Guillemin
and K. Okikiolu. They have computed the second term in a Szego type expansion
on a Zoll manifold of an arbitrary dimension. In the present work we compute
the third asymptotic term in any dimension. In the case of dimension 2, our
formula gives the above mentioned expression for the Szego-redularized
determinant of a zeroth order PsDO. The proof uses a new combinatorial
identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This
identity is related to the distribution of the maximum of a random walk with
i.i.d. steps on the real line. The proof of this combinatorial identity
together with historical remarks and a discussion of probabilistic and
algebraic connections has been published separately.Comment: 39 pages, full version, submitte
Characterization of n-rectifiability in terms of Jones' square function: Part II
We show that a Radon measure in which is absolutely
continuous with respect to the -dimensional Hausdorff measure is
-rectifiable if the so called Jones' square function is finite -almost
everywhere. The converse of this result is proven in a companion paper by the
second author, and hence these two results give a classification of all
-rectifiable measures which are absolutely continuous with respect to
. Further, in this paper we also investigate the relationship between
the Jones' square function and the so called Menger curvature of a measure with
linear growth.Comment: A corollary regarding analytic capacity and a few new references have
been adde
An Analyst's Traveling Salesman Theorem For Sets Of Dimension Larger Than One
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable
curves in the plane via a multiscale sum of -numbers. These
-numbers are geometric quantities measuring how far a given set deviates
from a best fitting line at each scale and location. Jones' result is a
quantitative way of saying that a curve is rectifiable if and only if it has a
tangent at almost every point. Moreover, computing this square sum for a curve
returns the length of the curve up to multiplicative constant. K. Okikiolu
extended his result from subsets of the plane to subsets of Euclidean space. G.
David and S. Semmes extended the discussion to include sets of (integer)
dimension larger than one, under the assumption of Ahlfors regularity and using
a variant of Jones' numbers. In this paper we give a version of P.
Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean
space. We estimate the -dimensional Hausdorff measure of a set in terms of
an analogous sum of -type numbers. There is no assumption of Ahlfors
regularity, but rather, only of a lower bound on the Hausdorff content. We
adapt David and Semmes' version of Jones' -numbers by redefining them
using a Choquet integral. A key tool in the proof is G. David and T. Toro's
parametrization of Reifenberg flat sets (with holes).Comment: Corrected more typos. There are still several typos and small
mistakes in the published version of the paper, so the authors will maintain
an up-to-date version on their webpages as we continue to correct the
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A Negative Mass Theorem for Surfaces of Positive Genus
Let M be a closed surface. For a metric g on M, denote the Laplace-Beltrami operator by Δ = Δ
g
. We define trace
, where dA is the area element for g and m(p) is the Robin constant at the point
, that is the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off. Since trace Δ−1 can also be obtained by regularization of the spectral zeta function, it is a spectral invariant. Heuristically it represents the sum of squares of the wavelengths of the surface. We define the Δ-mass of (M, g) to equal
, where
is the Laplacian on the round sphere of area A. This is an analog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus, the minimum of the Δ-mass on each conformal class is negative and attained by a smooth metric. For this minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality