Extremal Riemann Surfaces: From the Proceedings of the Ams by J. R. Quine, Peter Sarnak

By J. R. Quine, Peter Sarnak

This quantity is an outgrowth of the AMS certain consultation on Extremal Riemann Surfaces held on the Joint arithmetic assembly in San Francisco, January 1995. The booklet bargains with quite a few extremal difficulties relating to Riemann surfaces. a few papers take care of the identity of surfaces with longest systole (element of shortest nonzero size) for the size spectrum and the Jacobian. Parallels are interested in classical questions related to extremal lattices. different papers take care of maximizing or minimizing capabilities outlined by means of the spectrum comparable to the warmth kernel, the zeta functionality, and the determinant of the Laplacian, a few from the perspective of selecting an extremal metric.There are discussions of Hurwitz surfaces and surfaces with huge cyclic teams of automorphisms. additionally mentioned are surfaces that are common applicants for fixing extremal difficulties akin to triangular, modular, and mathematics surfaces, and curves in a variety of workforce theoretically outlined curve households. different allied issues are theta identities, quadratic classes of Abelian differentials, Teichmuller disks, binary quadratic kinds, and spectral asymptotics of degenerating hyperbolic 3 manifolds. This quantity: comprises papers by way of a number of the optimal specialists on Riemann surfaces; outlines attention-grabbing connections among Riemann surfaces and parallel fields; follows up on investigations of Sarnak touching on connections among the idea of utmost lattices and Jacobians of Riemann surfaces; and, includes papers on quite a few subject matters on the subject of Riemann surfaces

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Extra info for Extremal Riemann Surfaces: From the Proceedings of the Ams Special Session With Related Papers January 4-5, 1995 San Francisco, California

Example text

In this case, we define the generalized trace Tr(f ) of f by putting Tr(f ) = tr(f ). 1. Let f : E → E be an endomorphism. If dim E < ∞, then Tr(f ) = tr(f ). For the proof, see [22]. Let f = {fq } be an endomorphism of degree zero of a graded vector space E = {Eq }. We say that f is a Leray endomorphism if the graded vector space E = {Eq } is of finite type. For such an f , we define the (generalized) Lefschetz number (f ) of f by putting (f ) = (−1)q Tr(fq ). 2. , f = {fq } and fq : Eq → Eq is a linear map.

2. 3), where J is a given real interRn is a u-Carathéodory map and S is a subset of ACloc (J, Rn ). 1. 1 hold, with the convexity of the set Q, and (iv) if ∂Q × [0, 1] ⊃ {(qj , λj )} converges to (q, λ) ∈ ∂Q × [0, 1], q ∈ T (q, λ), then there exists j0 ∈ N such that, for every j j0 , and xj ∈ T (qj , λj ), we have xj ∈ Q. 3) has a solution. 3. 2. 2 that the linearized problems are uniquely solvable. 4) (corresponding to (x, λ)), then xj belongs to Q, for j sufficiently large. 3). 2 will be applied.

T1 This implies an equicontinuity of all x ∈ T (Q). We show that the set T is closed (cf. 1). Let T ⊃ {(qk , xk )} → (q, x). Let K be an arbitrary compact interval such that α is integrable on K. 1. Thus, there exists a subsequence (denoted just the same) {xk }, uniformly convergent to x on K (because the limit is unique) and such that {x˙k } weakly converges to x˙ in L1 . Therefore, x˙ belongs to the weak closure of the set conv{x˙m | m k}, for every k 1. By the mentioned Mazur theorem, x˙ also belongs to the strong closure of this set.

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