Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription

The manifestly-covariant Hamiltonian structure of classical General Relativity is shown to be associated with a path-integral synchronous Hamilton variational principle for the Einstein field equations. A realization of the same variational principle in both unconstrained and constrained forms is provided. As a consequence, the cosmological constant is found to be identified with a Lagrange multiplier associated with the normalization constraint for the extremal metric tensor. In particular, it is proved that the same Lagrange multiplier identifies a 4-scalar gauge function generally dependent on an invariant proper-time parameter s. Such a result is shown to be consistent with the prediction of the cosmological constant based on the theory of manifestly-covariant quantum gravity.

Numerical Approximations to Extremal Toric Kähler Metrics with Arbitrary Kähler Class

We develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on , which is conformal to an Einstein metric (the Chen–LeBrun–Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric that gives numerical evidence that the Einstein metric is conformally unstable under Ricci flow.

Optimal systolic inequalities on Finsler Möbius bands

We prove optimal systolic inequalities on Finsler Möbius bands relating the systole and the height of the Möbius band to its Holmes–Thompson volume. We also establish an optimal systolic inequality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric families both on the Möbius band and the Klein bottle are also presented.

COMPACT NON-ORIENTABLE SURFACES OF GENUS 5 WITH EXTREMAL METRIC DISCS

A compact hyperbolic surface of genus g is called an extremal surface if it admits an extremal disc, a disc of the largest radius determined by g. Our problem is to find how many extremal discs are embedded in non-orientable extremal surfaces. It is known that non-orientable extremal surfaces of genus g > 6 contain exactly one extremal disc and that of genus 3 or 4 contain at most two. In the present paper we shall give all the non-orientable extremal surfaces of genus 5, and find the locations of all extremal discs in those surfaces. As a consequence, non-orientable extremal surfaces of genus 5 contain at most two extremal discs.