BOUNDARY REGULARITY FOR WEAK HEAT FLOWS

61. IntroductionLet (M, g) be a compact smooth foemannian manifOld of dimension n with C2 boundary0M, and (N, h) be a smooth compact Riemannian manifolds of dimension k. Assume that(N, h) without boundary is isometrically embedded into the Euclidean space (Rm, (., .)).We assume that Sobolev spaceHl (M, N) = {u E Hl (M; R',.)lu(x) E N for a.e.x E M}and for every u E H1 (M; N), define the energy of u,E(u) = / lVuI'dv, (1.1)j. lVuI'dv, (1.1)where in local coordinate 1VuI' = g"pff 3, …     

Generalized Hopfian Property,a Minimal Haken Manifold,and Epimorphisms Between 3-Manifold Groups

We address the question that if π₁-surjective maps between closed aspherical 3-manifolds have the same rank on π₁ they must be of non-zero degree. The positive answer is proved for Seifert manifolds, which is used in constructing the first known example of minimal Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds. Many examples are given. Received September 16, 1999, Revised August 31, 2000, Accepted March 29, 2001.     

On Birational Maps and Jacobian Matrices

One is concerned with Cremona-like transformations, i.e., rational maps from ⁿ to ^m that are birational onto the image Y ^m and, moreover, the inverse map from Y to ⁿ lifts to ^m . We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.     

Degree-one maps between hyperbolic 3-manifolds with the same volume limit

Suppose that are degree-one maps between closed hyperbolic 3-manifolds with

Then, our main theorem, Theorem 2, shows that, for all but finitely many , is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston's rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.

     

Iterative Schemes for Solving Mixed Variational-Like Inequalities

In the present paper, we introduce the concept of -cocoercivity of a map and develop some iterative schemes for finding the approximate solutions of mixed variational-like inequalities. We use the concept of -cocoercivity to prove the convergence of the approximate solutions to the exact solution of mixed variational-like inequalities.     

On homology of real algebraic varieties

Let be a commutative ring with unity and an -oriented compact nonsingular real algebraic variety of dimension . If is any nonsingular complexification of , then the kernel, which we will denote by , of the induced homomorphism is independent of the complexification. In this work, we study and give some of its applications.

     

Fundamental groups of complements of plane curves and symplectic invariants

Introducing the notion of stabilized fundamental group for the complement of a branch curve in , we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these invariants and formulate conjectures supported by calculations on new examples.     

On Fourier algebra homomorphisms

Let G be a locally compact group and let B(G) be the dual space of C∗(G), the group C∗ algebra of G. The Fourier algebra A(G) is the closed ideal of B(G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism we show that it can be described, in terms of a piecewise affine map with Y in the coset ring of H, as follows