Stationary and stable varifolds with singularities

1. Problem Statement and Context

The paper addresses a fundamental question in Geometric Measure Theory regarding the structure of the singular set of stationary integral varifolds.

• Background: Following Allard's celebrated regularity theorem, it is known that stationary integral varifolds are smooth almost everywhere. However, the optimal dimension and structure of the singular set remain open problems, particularly for varifolds with multiplicity $\ge 2$.
• The Obstruction: A major obstruction to proving bounds on the size of the singular set is the potential accumulation of topology (e.g., handles or necks) on the smooth part of the varifold converging to a flat singular point.
• The Goal: The authors aim to construct explicit counterexamples demonstrating that such topological accumulation can occur when the ambient metric has low regularity ($C^{1,\alpha}$). Specifically, they seek to construct:
  1. A stationary minimal surface with an isolated singularity of multiplicity 2 and infinite genus (Theorem 1.1).
  2. A stationary and stable minimal surface with an isolated singularity of multiplicity 2, where the singularity is approached by a sequence of disjoint "floating disks" (Theorem 1.2).

2. Methodology

The construction relies on a sophisticated iterative gluing technique combined with metric perturbation.

A. The Models

The authors define three geometric models in $\mathbb{R}^{m+1}$:

1. The Base Model: The union of two minimal graphs ($u$ and $-u$) tangent at the origin. In 2D, this is constructed using Weierstrass data involving a holomorphic function $h(z)$ such that the resulting surface has a flat tangent cone of multiplicity 2 at the origin. In higher dimensions, they use $N$-homogeneous harmonic polynomials to ensure transversal intersections away from the origin.
2. The Catenoid (for Theorem 1.1): A sequence of catenoidal necks connecting the two sheets of the base model. As the necks shrink towards the origin, they introduce infinite topology (genus).
3. The Floating Disk (for Theorem 1.2): A more complex local model consisting of two catenoidal pieces and a flat disk meeting at a common circle with equal angles ($2\pi/3$). This forms a cycle mod 3 and is locally area-minimizing (stable) outside the origin.

B. The Gluing Strategy

The core of the proof involves inserting these local models (catenoids or floating disks) into the base model at a sequence of shrinking scales $r_k \to 0$.

• Partition of Unity: They use a smooth partition of unity to glue the base graph $u$ to the local model (catenoid or floating disk) in an annular region.
• Calibration Forms: To ensure the resulting surface remains stationary (and stable for Theorem 1.2) after gluing, they cannot simply use the Euclidean metric. Instead, they construct a calibration form $\omega$ in a tubular neighborhood of the glued surface.
  • $\omega$ is a closed form ($d\omega = 0$).
  • $\omega$ restricts to the volume form on the surface ($\omega|_S = d\text{vol}_S$).
  • The existence of such a form implies the surface is locally area-minimizing (or stationary) with respect to a specific metric.

C. Metric Perturbation

To make the calibration form valid, the authors perturb the Euclidean metric $g = \lambda^2 \delta$.

• They define a conformal factor $\lambda$ such that the comass of $\omega$ with respect to $g$ is 1.
• Regularity Analysis: The critical technical challenge is controlling the regularity of $\lambda$. The authors prove that while the metric is smooth away from the origin, it is only $C^{1,\alpha}$ at the origin.
  • The loss of higher regularity is attributed to the derivative estimates of the base model near the singularity, which are not "flat enough" to support a $C^{2,\alpha}$ metric when combined with the infinite sequence of surgeries.
  • They show that for any $\alpha \in [0, 1)$, they can choose parameters (specifically the degree $N$ of the harmonic polynomial) to achieve $C^{1,\alpha}$ regularity.

D. Iterative Construction

The construction is performed iteratively:

1. Start with the base model.
2. At scale $r_k$, glue in a local model (catenoid or floating disk).
3. Perturb the metric in the gluing region to preserve stationarity/stability.
4. Repeat for a sequence $r_k \to 0$ such that the total mass of the added boundaries (for floating disks) is finite.
5. Take the limit to obtain the final varifold $V$ and current $T$.

3. Key Results

Theorem 1.1 (Stationary with Infinite Genus)

• Existence: For every $\alpha \in [0, 1)$, there exists a $C^{1,\alpha}$ metric $g$ and an $m$-dimensional integral current $T$ stationary with respect to $g$.
• Properties:
  • $T$ corresponds to a minimal immersed surface $\Sigma$ in $B_1(0) \setminus \{0\}$.
  • The tangent cone at the origin is a flat plane with multiplicity 2 ($2 \llbracket \mathbb{R}^m \times \{0\} \rrbracket$).
  • Infinite Genus: The genus of $\Sigma \cap B_r(0) \setminus \{0\}$ is infinite for every $r > 0$.
• Implication: This provides a negative answer to the question of whether a stationary varifold with a double-plane tangent cone must be a union of two minimal graphs (as suggested by recent results under stronger topological assumptions).

Theorem 1.2 (Stationary and Stable)

• Existence: For every $\alpha \in [0, 1)$, there exists a $C^{1,\alpha}$ metric $g$ and a stationary, stable integral varifold $V$.
• Properties:
  • $V$ is supported on an immersed stable minimal surface $\Sigma$ with a sequence of disjoint embeddings $\gamma_k$ (the "floating disks") accumulating at the origin.
  • The tangent cone at 0 is a flat plane with multiplicity 2.
  • The surface consists of three components meeting at equal angles along the $\gamma_k$.
• Significance: This example shows that even stability (a stronger condition than stationarity) does not prevent the accumulation of topology to a singular flat point if the metric regularity is limited to $C^{1,\alpha}$. It suggests that the $H^{n-2}$ finiteness assumption in recent regularity theorems (e.g., Bellettini [3]) is sharp, as $H^{n-1}$ finiteness is insufficient.

4. Technical Significance and Contributions

1. Optimality of Regularity Assumptions: The paper demonstrates that the $C^{1,\alpha}$ regularity of the ambient metric is a critical threshold. In $C^{1,\alpha}$ metrics, the "topological obstruction" to regularity (accumulation of handles/disks) is realizable. This contrasts with higher regularity settings where such pathologies might be ruled out.
2. Counterexamples to Conjectures:
   • It refutes the idea that a double-plane tangent cone implies the surface is a union of two graphs (Theorem 1.1).
   • It challenges the sufficiency of $H^{n-1}$-finiteness for singular sets in stable varifolds, supporting the necessity of $H^{n-2}$ or capacity conditions (Theorem 1.2).
3. Calibration Technique: The authors develop a robust method for constructing calibration forms in tubular neighborhoods of glued surfaces. This allows them to control the first variation (stationarity) and second variation (stability) despite the complex geometry of the singular set.
4. Metric Construction: The construction of the metric is explicit and geometric. The authors show that the metric is smooth outside the origin, geodesically complete, and has a unique geodesic spray, making the singularity "mild" in a geometric sense, yet sufficient to break higher-order regularity of the varifold.

5. Conclusion

The paper establishes that in the context of $C^{1,\alpha}$ metrics, the singular set of stationary and stable varifolds can be far more complex than previously hoped, specifically allowing for the accumulation of infinite topology (genus) or disjoint singular components (floating disks) at a single point of multiplicity 2. These results highlight the delicate interplay between the regularity of the ambient metric and the structural regularity of minimal surfaces, setting a boundary for what can be proven in the theory of varifolds.