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Volume growth via real Lagrangians in Milnor fibers of Brieskorn polynomials

Joontae Kim, Myeonggi Kwon

Abstract

In this paper we study the volume growth in the component of fibered twists in Milnor fibers of Brieskorn polynomials. We obtain a uniform lower bound of the volume growth for a class of Brieskorn polynomials using a Smith inequality for involutions in wrapped Floer homology. To this end, we investigate a family of real Lagrangians in those Milnor fibers whose topology can be systematically described in terms of the join construction.

Volume growth via real Lagrangians in Milnor fibers of Brieskorn polynomials

Abstract

In this paper we study the volume growth in the component of fibered twists in Milnor fibers of Brieskorn polynomials. We obtain a uniform lower bound of the volume growth for a class of Brieskorn polynomials using a Smith inequality for involutions in wrapped Floer homology. To this end, we investigate a family of real Lagrangians in those Milnor fibers whose topology can be systematically described in terms of the join construction.
Paper Structure (17 sections, 17 theorems, 77 equations, 2 figures)

This paper contains 17 sections, 17 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Growth rate via Smith inequality
  3. Growth rate of wrapped Floer homology
  4. Wrapped Floer homology
  5. Action filtration
  6. Linear growth rate
  7. Smith inequality
  8. Stably trivial normal structures
  9. Smith inequality in filtered wrapped Floer homology
  10. Real Lagrangians in Milnor fibers of Brieskorn polynomials
  11. Real Lagrangians
  12. Topology of real Lagrangians
  13. Volume growth of fibered twists
  14. Fibered twists
  15. Growth rate of the real Lagrangians
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $V(\mathbf{a}) = V(a_0, a_1, \dots, a_n)$ be a Milnor fiber of a Brieskorn polynomial with $n \geq 3$ such that $a_j = 2$ for at least three $a_j$'s and at most one $a_j$ is odd. Let $\vartheta$ be a fibered twist on $V(\mathbf{a})$. Then $s_n(\varphi) \geq 1$ for any compactly supported symplec

Figures (2)

  • Figure 1: Reflections on ${\mathbb{C}}$
  • Figure 2: The geometric setup of the Lagrangian tomograph

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 34 more