Ricci flow and Poincare conjecture - collection of research

Read Online or Download Ricci flow and Poincare conjecture - collection of research papers PDF

Similar geometry and topology books

Real Methods in Complex and CR Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, June 30 - July 6, 2002

The geometry of genuine submanifolds in advanced manifolds and the research in their mappings belong to the main complicated streams of up to date arithmetic. during this zone converge the concepts of varied and complex mathematical fields equivalent to P. D. E. 's, boundary price difficulties, caused equations, analytic discs in symplectic areas, complicated dynamics.

Designing fair curves and surfaces: shape quality in geometric modeling and computer-aided design

This cutting-edge examine of the options used for designing curves and surfaces for computer-aided layout functions specializes in the primary that reasonable shapes are continuously freed from unessential positive factors and are basic in layout. The authors outline equity mathematically, reveal how newly built curve and floor schemes warrantly equity, and support the consumer in opting for and removal form aberrations in a floor version with out destroying the important form features of the version.

Additional info for Ricci flow and Poincare conjecture - collection of research papers

Sample text

10* The earlier work on ancient solutions and all that can be found in [H 4, §16 − 22, 25, 26]. 1 Let φ be a decreasing function of one variable, tending to zero at infinity. A solution to the Ricci flow is said to have φ-almost nonnegative curvature if it satisfies Rm(x, t) ≥ −φ(R(x, t))R(x, t) for each (x, t). Theorem. Given ǫ > 0, κ > 0 and a function φ as above, one can find r0 > 0 with the following property. 1. Proof. An argument by contradiction. Take a sequence of r0 converging to zero, and consider the solutions gij (t), such that the conclusion does not hold for some (x0 , t0 ); moreover, by tampering with the condition t0 ≥ 1 a little bit, choose among all such (x0 , t0 ), in the solution under consideration, the one with nearly the smallest curvature Q.

Slightly abusing notation, we’ll drop the indices α, β when we consider an individual solution. Let t¯ be the first time when the assumption is violated at some point x¯; clearly such time exists, because it is an open condition. 2 we have uniform κ-noncollapsing on [0, t¯]. 2 are also valid on [0, t¯]; moreover, since h << r, it follows from Claim 1 that the solution is defined on the whole parabolic neighborhood indicated there in case R(x0 , t0 ) ≤ r−2 . Scale our solution about (¯ x, t¯) with factor R(¯ x, t¯) ≥ r−2 and take a limit for ¯ subsequences of α, β → ∞.

The point is to make h arbitrarily small while keeping r bounded away from zero. Notation and terminology B(x, t, r) denotes the open metric ball of radius r, with respect to the metric at time t, centered at x. P (x, t, r, △t) denotes a parabolic neighborhood, that is the set of all points (x′ , t′ ) with x′ ∈ B(x, t, r) and t′ ∈ [t, t + △t] or t′ ∈ [t + △t, t], depending on the sign of △t. A ball B(x, t, ǫ−1 r) is called an ǫ-neck, if, after scaling the metric with factor −2 r , it is ǫ-close to the standard neck S2 × I, with the product metric, where S2 has constant scalar curvature one, and I has length 2ǫ−1 ; here ǫ-close refers to C N topology, with N > ǫ−1 .

Download PDF sample

Rated 4.63 of 5 – based on 5 votes