Remarks on the extension of the Ricci flow

We present two new conditions to extend the Ricci flow on a compact manifold over a finite time, which are improvements of some known extension theorems.Comment: 9 pages, to appear in Journal of Geometric Analysi

Rigidity around Poisson Submanifolds

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem; another interesting case corresponds to spheres inside duals of compact semisimple Lie algebras, our result can be used to fully compute the resulting Poisson moduli space.Comment: 43 pages, v3: published versio

Normalized Ricci flow on Riemann surfaces and determinants of Laplacian

In this note we give a simple proof of the fact that the determinant of Laplace operator in smooth metric over compact Riemann surfaces of arbitrary genus $g$ monotonously grows under the normalized Ricci flow. Together with results of Hamilton that under the action of the normalized Ricci flow the smooth metric tends asymptotically to metric of constant curvature for $g\geq 1$, this leads to a simple proof of Osgood-Phillips-Sarnak theorem stating that that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of Laplace operator is maximal on metric of constant curvatute.Comment: a reference to paper math.DG/9904048 by W.Mueller and K.Wendland where the main theorem of this paper was proved a few years earlier is adde

Ricci flows with unbounded curvature

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).Comment: 12 pages, 1 figure; updated reference

The Ricci flow on noncommutative two-tori

In this paper we construct a version of Ricci flow for noncommutative 2-tori, based on a spectral formulation in terms of the eigenvalues and eigenfunction of the Laplacian and recent results on the Gauss-Bonnet theorem for noncommutative tori.Comment: 18 pages, LaTe

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

Shakedown analysis for rolling and sliding contact problems

There is a range of problems where repeated rolling or sliding contact occurs. For such problems shakedown and limit analyses provides significant advantages over other forms of analysis when a global understanding of deformation behaviour is required. In this paper, a recently developed numerical method. Ponter and Engelhardt (2000) and Chen and Ponter (2001), for 3-D shakedown analyses is used to solve the rolling and sliding point contact problem previously considered by Ponter, Hearle and Johnson (1985) for a moving Herzian contact, with friction, over a half space. The method is an upper bound programming method, the Linear Matching Method, which provides a sequence of reducing upper bounds that converges to the least upper bound associated with a finite element mesh and may be implemented within a standard commercial finite element code. The solutions given in Ponter, Hearle and Johnson (1985) for circular contacts are reproduced and extended to the case when the frictional contact stresses are at an angle to the direction of travel. Solutions for the case where the contact region is elliptic are also given

On the geometry of the space of fibrations

We study geometrical aspects of the space of fibrations between two given manifolds M and B, from the point of view of Frechet geometry. As a first result, we show that any connected component of this space is the base space of a Frechet-smooth principal bundle with the identity component of the group of diffeomorphisms of M as total space. Second, we prove that the space of fibrations is also itself the total space of a smooth Frechet principal bundle with structure group the group of diffeomorphisms of the base B.Comment: 18 pages, 5 figure

The K\"ahler-Ricci flow with positive bisectional curvature

We show that the K\"ahler-Ricci flow on a manifold with positive first Chern class converges to a K\"ahler-Einstein metric assuming positive bisectional curvature and certain stability conditions.Comment: 15 page

The Degasperis-Procesi equation as a non-metric Euler equation

In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of $b$-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\'echet space of all smooth functions on the circle.Comment: 17 page