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Uniqueness of tangent cones for calibrated 2-cycles

David Pumberger, Tristan Riviere

TL;DR

The paper proves the uniqueness of tangent cones for $2$-dimensional calibrated cycles and provides a quantitative rate of convergence for the mass of blow-ups toward the limiting density; it also extends these methods to $J$-holomorphic maps between almost complex manifolds, establishing uniqueness of tangent maps. The authors develop a blow-up/compactness framework built around almost monotonicity formulas, a Hopf-type projection to $\mathbb{CP}^{m-1}$, facilitating comparison of tangent cones and deriving decay estimates. For calibrated $2$-cycles, they show uniqueness of tangent cones by analyzing convex combinations of calibrated simple 2-vectors and using the projection to deduce equality, and derive a rate $M(\pi_* C B_r(x_0)) \le C r^\gamma$ with $\gamma \in (0,1)$. The analysis is first carried out in the standard complex structure and then extended to perturbations $J$ near $J_0$, yielding an almost monotonicity formula, density existence, energy-decay, and uniqueness of tangent maps in $W^{1,2}$.

Abstract

We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex manifolds and deduce the uniqueness of their tangent maps.

Uniqueness of tangent cones for calibrated 2-cycles

TL;DR

The paper proves the uniqueness of tangent cones for -dimensional calibrated cycles and provides a quantitative rate of convergence for the mass of blow-ups toward the limiting density; it also extends these methods to -holomorphic maps between almost complex manifolds, establishing uniqueness of tangent maps. The authors develop a blow-up/compactness framework built around almost monotonicity formulas, a Hopf-type projection to , facilitating comparison of tangent cones and deriving decay estimates. For calibrated -cycles, they show uniqueness of tangent cones by analyzing convex combinations of calibrated simple 2-vectors and using the projection to deduce equality, and derive a rate with . The analysis is first carried out in the standard complex structure and then extended to perturbations near , yielding an almost monotonicity formula, density existence, energy-decay, and uniqueness of tangent maps in .

Abstract

We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex manifolds and deduce the uniqueness of their tangent maps.
Paper Structure (14 sections, 16 theorems, 139 equations)

This paper contains 14 sections, 16 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Proof of theorem \ref{['J-maps']}
  3. The case of the standard complex structure
  4. Perturbing the proof
  5. A second order elliptic equation for $u$
  6. Monotonicity formula for the energy of $u$
  7. Obtaining a rate of convergence for $u$
  8. Uniqueness of tangent maps
  9. Proofs of theorems \ref{['uniqueness']} and \ref{['rate']}
  10. Preliminaries
  11. Proof of theorem \ref{['uniqueness']} for calibrated $J_0$-holomorphic $2$-cycles
  12. Obtaining a rate for integer rectifiable $2$-cycles calibrated by $\omega _0$
  13. Uniqueness of tangent cones for calibrated $2$-cycles
  14. Proof of theorem \ref{['rate']}

Key Result

Theorem 1

Let u:M\rightarrow N be a locally approximable J-holomorphic map from an almost complex manifold (M, J) into a tamed compact symplectic manifold (N, J_N, \omega _N), such that u\in W^{1,2}(M,N). Given any x_0\in M there exist C_1, C_2>0 and \gamma\in (0,1] independent of u such that where \Theta _u(x_0):=\lim _{\rho\rightarrow 0}\frac{1}{\rho ^{2n-2}}\int _{B_\rho(x_0)} |\nabla u|^2.

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Lemma 1
  • ...and 7 more