Calibrations for the Sasaki volume on odd spheres and the no-gap problem

TL;DR

This paper establishes a universal lower bound for the Sasaki volume functional on odd spheres using calibration techniques, characterizes equality cases, and proves the absence of a Lavrentiev gap through explicit constructions and estimates.

Contribution

It introduces a calibration-based lower bound for the Sasaki volume on odd spheres, analyzes the structure of equality cases, and constructs explicit sequences to show no gap exists.

Findings

Universal calibrated lower bound for Sasaki volume on odd spheres.

Characterization of $ abla_V V=0$ and $ abla_X V= ext{const} imes X$ as rigidity conditions.

Proof of no Lavrentiev gap via explicit recovery sequences.

Abstract

For each odd sphere $S^n$ with $n=2m+1\ge 5$, we consider the Sasaki volume functional $\mathrm{Vol}^S(V)=\int_{S^n}\sqrt{\det(I+(\nabla V)^\top(\nabla V))}\,d\mathrm{vol}$ on smooth unit tangent vector fields $V$. Using the Gluck--Ziller calibration $\omega=a\wedge\Theta$ on the unit tangent bundle $E=UT S^n$ (extended to constant sectional curvature by Brito--Chac\'on--Naveira), we establish the universal calibrated lower bound $\mathrm{Vol}^S(V)\ge c(m;1)\,\mathrm{vol}(S^n)$, where $c(m;1)=4^m/\binom{2m}{m}$. In the relaxed (integral-current) setting, we show that the section-constrained stable mass in $E$ equals the calibration value and is attained by an $\omega$-calibrated mass-minimizing integral $n$-cycle in the section class. We also analyze the equality case on smooth graphs. If a smooth graph is $\omega$-calibrated on an open set, then it satisfies the rigidity system $\nabla_V V=0$ and $\nabla_X V=\lambda X$ for all $X\perp V$, hence is locally a radial distance-gradient field. In particular, for $m\ge 2$ there is no smooth unit field on $S^n$ whose graph is $\omega$-calibrated everywhere. Finally, we construct an explicit smooth recovery sequence (presented in detail for $S^5$ and then extended to all odd dimensions) and prove a uniform nonvanishing estimate for the polar-shell normalization in the patching construction. As a consequence, $\inf_V \mathrm{Vol}^S(V)=c(m;1)\,\mathrm{vol}(S^n)$, so there is no Lavrentiev gap.