201 research outputs found
The K\"ahler-Ricci flow on surfaces of positive Kodaira dimension
The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has
been the subject of intensive study over the last few decades, following Yau's
solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton
has become one of the most powerful tools in geometric analysis.
We study the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension one
and show that the flow collapses and converges to a unique canonical metric on
its canonical model. Such a canonical is a generalized K\"ahler-Einstein
metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric
classification for K\"aher surfaces with a numerical effective canonical line
bundle by the K\"ahler-Ricci flow. In general, we propose a program of finding
canonical metrics on canonical models of projective varieties of positive
Kodaira dimension
Education in IT Security: A Case Study in Banking Industry
The banking industry has been changing incessantlyand facing new combination of risks. Data protection andcorporate security is now one of the major issues in bankingindustry. As the rapid changing on technologies from time totime, the industry should be aware on new technologies in orderto protect information assets and prevent fraud activities. Thispaper begins with literature study of information security issuesand followed by focused-group interviews with five participantswithin the industry and survey analysis of “The global state ofInformation Security survey 2013” which published byPriceWaterhouseCoopers (PWC). Trends and questions werediscussed as well as possible solution. The study suggests that ITsecurity education should be made to different level of staffs suchas executives, professional and general staffs. Besides, thebanking industry should increase company-wide securityawareness and the importance of corporate security which keepthe information and physical assets secure and in a proper way
Del Pezzo surfaces with many symmetries
We classify smooth del Pezzo surfaces whose alpha-invariant of Tian is bigger
than one.Comment: 23 page
Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic
We develop an iterative method for finding solutions to the hermitian
Yang-Mills equation on stable holomorphic vector bundles, following ideas
recently developed by Donaldson. As illustrations, we construct numerically the
hermitian Einstein metrics on the tangent bundle and a rank three vector bundle
on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank
three vector bundle on the Fermat quintic.Comment: 25 pages, 2 figure
A Spectral Bernstein Theorem
We study the spectrum of the Laplace operator of a complete minimal properly
immersed hypersurface in . (1) Under a volume growth condition on
extrinsic balls and a condition on the unit normal at infinity, we prove that
has only essential spectrum consisting of the half line .
This is the case when , where
is the extrinsic distance to a point of and are the
principal curvatures. (2) If the satisfy the decay conditions
, and strict inequality is achieved at some point
, then there are no eigenvalues. We apply these results to minimal
graphic and multigraphic hypersurfaces.Comment: 16 pages. v2. Final version: minor revisions, we add Proposition 3.2.
Accepted for publication in the Annali di Matematica Pura ed Applicata, on
the 05/03/201
Existence of Ricci flows of incomplete surfaces
We prove a general existence result for instantaneously complete Ricci flows
starting at an arbitrary Riemannian surface which may be incomplete and may
have unbounded curvature. We give an explicit formula for the maximal existence
time, and describe the asymptotic behaviour in most cases.Comment: 20 pages; updated to reflect galley proof correction
Manifolds with 1/4-pinched flag curvature
We say that a nonnegatively curved manifold has quarter pinched flag
curvature if for any two planes which intersect in a line the ratio of their
sectional curvature is bounded above by 4. We show that these manifolds have
nonnegative complex sectional curvature. By combining with a theorem of Brendle
and Schoen it follows that any positively curved manifold with strictly quarter
pinched flag curvature must be a space form. This in turn generalizes a result
of Andrews and Nguyen in dimension 4. For odd dimensional manifolds we obtain
results for the case that the flag curvature is pinched with some constant
below one quarter, one of which generalizes a recent work of Petersen and Tao
Extremal Bundles on Calabi-Yau Threefolds
We study constructions of stable holomorphic vector bundles on Calabi–Yau threefolds, especially those with exact anomaly cancellation which we call extremal. By going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds. We then introduce a general Hartshorne–Serre construction and use it to find extremal bundles of general ranks and study their stability, as well as computing their Chern numbers. Based on both existing and our new constructions, we revisit the DRY conjecture for the existence of stable sheaves on Calabi–threefolds, and provide theoretical and numerical evidence for its correctness. Our construction can be easily generalized to bundles with no extremal conditions imposed
On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces
We show that the spectrum of a complete submanifold properly immersed into a
ball of a Riemannian manifold is discrete, provided the norm of the mean
curvature vector is sufficiently small. In particular, the spectrum of a
complete minimal surface properly immersed into a ball of is
discrete. This gives a positive answer to a question of Yau.Comment: This article is an improvement of an earlier version titled On the
spectrum of Martin-Morales minimal surfaces. 7 page
Hyperholomorpic connections on coherent sheaves and stability
Let be a hyperkaehler manifold, and a torsion-free and reflexive
coherent sheaf on . Assume that (outside of its singularities) admits a
connection with a curvature which is invariant under the standard SU(2)-action
on 2-forms. If the curvature is square-integrable, then is stable and its
singularities are hyperkaehler subvarieties in . Such sheaves (called
hyperholomorphic sheaves) are well understood. In the present paper, we study
sheaves admitting a connection with SU(2)-invariant curvature which is not
necessarily square-integrable. This situation arises often, for instance, when
one deals with higher direct images of holomorphic bundles. We show that such
sheaves are stable.Comment: 37 pages, version 11, reference updated, corrected many minor errors
and typos found by the refere
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