By Leon Simon

ISBN-10: 376435397X

ISBN-13: 9783764353971

The purpose of those lecture notes is to offer an primarily self-contained creation to the elemental regularity idea for strength minimizing maps, together with fresh advancements about the constitution of the singular set and asymptotics on method of the singular set. really good wisdom in partial differential equations or the geometric calculus of adaptations is no longer required. as a substitute, an excellent normal heritage in mathematical research will be sufficient instruction.

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**Theorems on regularity and singularity of energy minimizing - download pdf or read online**

The purpose of those lecture notes is to offer an basically self-contained advent to the fundamental regularity concept for strength minimizing maps, together with contemporary advancements about the constitution of the singular set and asymptotics on method of the singular set. really good wisdom in partial differential equations or the geometric calculus of adaptations is no longer required.

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**Extra resources for Theorems on regularity and singularity of energy minimizing maps**

**Example text**

A compact, reduced, connected curve C on a nonsingular surface X is called exceptional, if there is a bimeromorphic map 7r : X --+ Y, such that there is an open neighbourhood U of C in X , a point y E I/, and a neighbourhood V of y in Y with the property that 7r : U \ C --+ V \ {y} is biholomorphie and 7r(C) = y. We shall say also in this case that C is contracted to y. CN) is negative definite. e. 3 Classification and examples of surfaces 39 frequently called (-1)-curves. In fact, the contraction of a (-1)-curve is the inverse construction of a a-process.

E. F is a free subgroup of C 2 of rank 4). 3 Let F be defined by the vectors Vl, v2, v3, v 4 linearly independent over IR, F = 7/Vl + 7/v2 + 7/v3 + 77v4 and let X = C2/F. The complex surface X is homeomorphic to (IR/7/) 4 and H2(X, 77) _~ 776 where the group H2(X, 77) is generated by the cycles Sij , 1 < i < j <_ 4, obtained as the images of the planes ]Rvi + IRvj by the natural projection (~2 _+ X. Let C C X be a curve on X. Taking a triangulation of the normalization of C we can consider the curve C in a natural way as a singular cycle on X.

3), the existence of a holomorphic structure in a given topological vector bundle is equivalent to the existence of an algebraic structure. Let us consider the case dim X = 3. If E is topological vector bundle of rank r > 3, then the Chern classes (Cl(E), c2(E), c3(E)) determine up to an isomorphism the bundle E. This fact follows by the above mentioned results and it was firstly proved in [Bc]. If r = rk(E) > 3, then E ~ E ' @ (r - 3)11, where E ' is a vector bundle of rank 3, uniquely determined (up to an isomorphism) by the isomorphism class of E.

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