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

Jonas Matuzas

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 Brito--Chacon--Naveira calibration $ω=a\wedgeΘ$ on the unit tangent bundle $E=UTS^{n}$, 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 $ω$-calibrated mass-minimizing integral $n$-cycle in the section class. We also analyze the equality case on smooth graphs. If a smooth graph is $ω$-calibrated on an open set, then it satisfies the rigidity system $\nabla_V V=0$ and $\nabla_X V=λX$ 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 $ω$-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.

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

Abstract

For each odd sphere with , we consider the Sasaki volume functional on smooth unit tangent vector fields . Using the Brito--Chacon--Naveira calibration on the unit tangent bundle , we establish the universal calibrated lower bound , where . In the relaxed (integral-current) setting, we show that the section-constrained stable mass in equals the calibration value and is attained by an -calibrated mass-minimizing integral -cycle in the section class. We also analyze the equality case on smooth graphs. If a smooth graph is -calibrated on an open set, then it satisfies the rigidity system and for all , hence is locally a radial distance-gradient field. In particular, for there is no smooth unit field on whose graph is -calibrated everywhere. Finally, we construct an explicit smooth recovery sequence (presented in detail for 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, , so there is no Lavrentiev gap.
Paper Structure (21 sections, 76 theorems, 775 equations)

This paper contains 21 sections, 76 theorems, 775 equations.

Table of Contents

  1. Introduction
  2. Geometry of UTS(n) as a Stiefel manifold and the Sasaki frame
  3. Basic identities on the round sphere
  4. An invariant Sasaki calibration and the polynomial Phi d
  5. The Theta-calibrated d-planes (linear algebra bottleneck)
  6. The BCN lower bound and a recovery-sequence ansatz (model case: S5)
  7. A smooth recovery sequence on S5
  8. General odd spheres S(2m+1)
  9. A rotation--based definition of $\gamma_k$ (odd spheres).
  10. Strong minimizers and the calibrated minimizer current
  11. Equality case on smooth graphs: b=0 and M=lambda Id
  12. Sphere rigidity: explicit center formula and radiality
  13. Globalization and pole behavior in E
  14. Final theorems (model case: S5)
  15. GMT tools used (precise statements)
  16. ...and 6 more sections

Key Result

Lemma 1.2

Let $V$ be a smooth unit vector field on $\mathbb S^n$ and let $s_V\colon \mathbb S^n\to E=\mathrm{UT}\mathbb S^n$ be the associated section $s_V(x)=(x,V(x))$. Define the graph current $T\Gamma_V:=(s_V)_\#\llbracket \mathbb S^n\rrbracket$. Then $T\Gamma_V\in\mathbf I_n(E)$, $\partial T\Gamma_V=0$, a In particular, $T\Gamma_V\in\mathcal{S}$.

Theorems & Definitions (170)

  • Definition 1.1: Section--constrained stable mass
  • Lemma 1.2: Graph currents satisfy the section constraint
  • proof
  • Definition 1.3: $n$--Jacobian of a linear map
  • Definition 1.4: Rectifiable sections and multiplicity-one
  • Lemma 2.1: Levi--Civita connection on the unit sphere in ambient coordinates
  • proof
  • Lemma 2.2: Distance function formulas on the unit sphere
  • proof
  • Lemma 2.3: Geodesic polar coordinates on the unit sphere
  • ...and 160 more