By V. Gusev, V. Litvinenko, A. Morkovich
Read or Download Problemas Matematicos - Geometria PDF
Similar geometry and topology books
From Geometry to Quantum Mechanics: In Honor of Hideki Omori
This quantity consists of invited expository articles via recognized mathematicians in differential geometry and mathematical physics which were prepared in occasion of Hideki Omori's contemporary retirement from Tokyo college of technological know-how and in honor of his primary contributions to those parts.
Designing fair curves and surfaces: shape quality in geometric modeling and computer-aided design
This cutting-edge learn of the thoughts used for designing curves and surfaces for computer-aided layout functions specializes in the primary that reasonable shapes are continually freed from unessential beneficial properties and are easy in layout. The authors outline equity mathematically, show how newly built curve and floor schemes warrantly equity, and support the person in picking out and removal form aberrations in a floor version with no destroying the imperative form features of the version.
Professor Peter Hilton is without doubt one of the top identified mathematicians of his iteration. He has released nearly three hundred books and papers on quite a few elements of topology and algebra. the current quantity is to have a good time the party of his 60th birthday. It starts off with a bibliography of his paintings, via reports of his contributions to topology and algebra.
Additional info for Problemas Matematicos - Geometria
Sample text
11) is n M (∆f − |∇f |2 )[τ (R + 2∆f − |∇f |2 ) + f ](4πτ )− 2 e−f dV n = M − (∆f − |∇f |2 )(2τ ∆f − τ |∇f |2 )(4πτ )− 2 e−f dV n M n (−∇i f )(∇i R)(4πτ )− 2 e−f dV M n 2 =τ M − |∇f |2 (4πτ )− 2 e−f dV + τ (−∇i f )(∇i (2∆f − |∇f | ))(4πτ )− 2 e−f dV n M ∆f (4πτ )− 2 e−f dV − 2τ n M ∇i f ∇j Rij (4πτ )− 2 e−f dV n = −2τ M (∇i f )(∇i ∆f − ∇f, ∇i ∇f )(4πτ )− 2 e−f dV M [(∇i ∇j f )Rij − ∇i f ∇j f Rij ](4πτ )− 2 e−f dV n + 2τ + 2τ M = −2τ M − 1 n gij (∇i ∇j f )(4πτ )− 2 e−f dV 2τ [(∇i f ∇j f − ∇i ∇j f )∇i ∇j f − Rij ∇i f ∇j f n − ∇i ∇j f ∇i f ∇j f ](4πτ )− 2 e−f dV n + 2τ M [(∇i ∇j f )Rij − ∇i f ∇j f Rij ](4πτ )− 2 e−f dV 1 n gij (∇i ∇j f )(4πτ )− 2 e−f dV 2τ M n 1 = 2τ (∇i ∇j f ) ∇i ∇j f + Rij − gij (4πτ )− 2 e−f dV.
Dt THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 205 On the other hand, for every t, − d 1 log V = dt V M RdV ≥ λ(gij (t)) by the definition of λ(gij (t)). It follows that on an expanding breather on [t1 , t2 ], ¯ ij (t)) = λ(gij (t))V n2 (gij (t)) < 0 λ(g for some t ∈ [t1 , t2 ]. Then by using statement (i), it implies ¯ ij (t1 )) < λ(g ¯ ij (t2 )) λ(g ¯ ij (t)) is invariant unless we are on an expanding gradient soliton. We also note that λ(g under diffeomorphism and scaling which implies ¯ ij (t1 )) = λ(g ¯ ij (t2 )).
In [60], Hamilton obtained the following Li-Yau estimate for the scalar curvature R(x, t). 2 (Hamilton [60]). Let gij (x, t) be a complete solution of the Ricci flow on a surface M . Assume the scalar curvature of the initial metric is bounded, nonnegative everywhere and positive somewhere. Then the scalar curvature R(x, t) satisfies the Li-Yau estimate ∂R |∇R|2 R − + ≥ 0. 4) Proof. By the above discussion, we know R(x, t) > 0 for t > 0. 4) is equivalent to ∂L 1 1 − |∇L|2 + = △L + R + ≥ 0. 5) Q= ∂L − |∇L|2 = △L + R.