A Pu–Bonnesen inequality

Katz, Mikhail G.; Sabourau, Stéphane

doi:10.1007/s00022-021-00579-2

J. Geom.

2021

DOI: 10.1007/s00022-021-00579-2

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A Pu–Bonnesen inequality

Mikhail G. Katz

Stéphane Sabourau

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Cited by 2 publications

(2 citation statements)

References 17 publications

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“…We have learned of work by Katz and Sabourau[24], dealing with systolic geometry of surfaces with Riemannian metrics of non-positive curvature. Regions without systolic geodesics play an important role in this work.…”

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confidence: 99%

Convex Programs for Minimal-Area Problems

Headrick

Zwiebach

2020

Commun. Math. Phys.

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The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the analogous minimal-area problem for homology classes of curves and, with the aid of calibrations and the max flow-min cut theorem, formulate it as a local convex program. We derive an equivalent dual program involving maximization of a concave functional. These two programs give new insights into the form of the minimal-area metric and are amenable to numerical solution. We explain how the homology problem can be modified to provide the solution to the original homotopy problem.

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confidence: 99%

Convex Programs for Minimal-Area Problems

Headrick

Zwiebach

2020

Commun. Math. Phys.

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“…This is in contrast to the metrics arising from Jenkins-Strebel quadratic differentials, since those metrics are locally flat and do not have line curvature 4. We have learned of work by Katz and Sabourau[23], dealing with systolic geometry of surfaces with Riemannian metrics of non-positive curvature. Regions without systolic geodesics play an important role in this work.…”

mentioning

confidence: 99%

Minimal-Area Metrics on the Swiss Cross and Punctured Torus

Headrick

Zwiebach

2020

Commun. Math. Phys.

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The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimalarea metric on the square torus with a square boundary, for various sizes of the boundary. For large enough boundary the problem is equivalent to the "Swiss cross" challenge posed by Strebel. We find that the metric is positively curved in the two-band region and flat in the single-band regions. For small boundary the metric develops a third band of geodesics wrapping around it, and has both regions of positive and negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram.

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A Pu–Bonnesen inequality

Cited by 2 publications

References 17 publications

Convex Programs for Minimal-Area Problems

Convex Programs for Minimal-Area Problems

Minimal-Area Metrics on the Swiss Cross and Punctured Torus

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