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Morse index of y-singular minimal surfaces

Elham Matinpour

Abstract

In this paper, we compute the Morse index of rotationally symmetric minimal Y-singular surfaces under assumption that the Y-singularities form a single circle. This computation is carried out by utilizing information from two simpler problems: the first deals with the fixed boundary problem on singularities, and the second focuses on the Dirichlet-to-Neumann map associated with the stability operator. Notably, our findings reveal that the index of the Y -catenoid is one.

Morse index of y-singular minimal surfaces

Abstract

In this paper, we compute the Morse index of rotationally symmetric minimal Y-singular surfaces under assumption that the Y-singularities form a single circle. This computation is carried out by utilizing information from two simpler problems: the first deals with the fixed boundary problem on singularities, and the second focuses on the Dirichlet-to-Neumann map associated with the stability operator. Notably, our findings reveal that the index of the Y -catenoid is one.

Paper Structure

This paper contains 15 sections, 14 theorems, 191 equations, 3 figures.

Table of Contents

  1. Introduction
  2. background and notation
  3. Basic Function Spaces
  4. Triple Junction Minimal Surfaces.
  5. Family of Y-noids
  6. index form evaluation
  7. Index Decomposition.
  8. General Index Decomposition
  9. Morse Index of the $Y$-noid family
  10. Index Form on the Example of $Y$-Catenoid
  11. Index Form on the Example of Pseudo $Y$-Catenoid
  12. Evaluating the Index of Other $Y$-Noids Family Members
  13. Assessment of the Index Form for Members in the Subfamily $\{ Y_{\alpha} \}_{\alpha \in (0,\frac{\pi}{6}) }$
  14. Assessment of the Index Form for Members in the Subfamily $\{ Y_{\alpha} \}_{\alpha \in (\frac{\pi}{6}, \frac{\pi}{3}) }$
  15. Index Form on the Surface $Y_{\frac{\pi}{6}}$

Key Result

Proposition 3.1

Fix $1\leq i\leq 3$. For any $f\in C^{\infty} (\Gamma_i)$ there is a unique function $u\in C^{\infty} (M_i)$ satisfying: $(1)\ u\vert_{\partial ' M_i}=0\ \text{and}\ u\vert_{\Gamma_i} =f;$$(2)\ J_i u \in \mathcal{K}_i,\ \text{i. e., if}\ \mathcal{K}_i=\{0\} \ \text{is trivial this is the usual Ja for this unique extension.

Figures (3)

  • Figure 1:
  • Figure 2: a. The $Y$-catenoid(on the left), b. The pseudo $Y$-catenoid(on the right)
  • Figure 3: An example of $Y$-noids family

Theorems & Definitions (37)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 27 more