The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric. For area π/3, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima.On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular 7-gon is the optimal n-gonal tile of area π/3 for 3 ≤ n ≤ 10. However, for n > 10, it is difficult to rule out non-convex n-gons that tile irregularly. Contents 1. Introduction 1 Methods 2 Acknowledgements 2 2. Definitions 3 3. Hyperbolic Geometry 4 4.

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In 1993 Foisy et al. [4] proved that the optimal Euclidean planar double bubble-the least-perimeter way to enclose and separate two given areas-is three circular arcs meeting at 120 degrees. We consider the plane with density r p , joining the surge of research on manifolds with density after their appearance in Perelman's 2006 proof of the Poincaré Conjecture. Dahlberg et al. [3] proved that the best single bubble in the plane with density r p is a circle through the origin. We conjecture that the best double bubble is the Euclidean solution with one of the vertices at the origin, for which we have verified equilibrium (first variation or "first derivative" zero). To prove the exterior of the minimizer connected, it would suffice to show that least perimeter is increasing as a function of the prescribed areas. We give the first direct proof of such monotonicity for the Euclidean case. Such arguments were important in the 2002 Annals proof [7] of the double bubble in Euclidean 3-space.

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Blaschke factorization allows us to write any holomorphic function F as a formal serieswhere a i ∈ C and B i is a Blaschke product. We introduce a more general variation of the canonical Blaschke product and study the resulting formal series. We prove that the series converges exponentially in the Dirichlet space given a suitable choice of parameters if F is a polynomial and we provide explicit conditions under which this convergence can occur. Finally, we derive analogous properties of Blaschke factorization using our new variable framework.2010 Mathematics Subject Classification. 30A10, 30B50.