A sharp quantitative stability result near infinitely concentrated minimisers

1. Problem Statement and Context

The Variational Problem:
The authors investigate the Dirichlet energy $E(v)$ for maps $v: (\Sigma, g_\Sigma) \to S^2$ from a closed orientable surface $\Sigma$ of positive genus ($\gamma \geq 1$) to the unit sphere. The energy is defined as:
$$E(v) = \frac{1}{2} \int_\Sigma |\nabla v|^2 d\text{vol}_{g_\Sigma}$$
It is well-known that for maps of degree $k$, the energy is bounded below by $4\pi k$. For degree $k=1$, the infimum is $4\pi$.

The Core Difficulty:

• Non-Existence of Classical Minimisers: Unlike the case where $\Sigma = S^2$ (where minimisers are Möbius transformations), the Riemann-Hurwitz formula implies that no degree 1 conformal map exists from a surface of positive genus to $S^2$. Thus, the set of classical minimisers is empty.
• Infinitely Concentrated Minimisers: The infimum is approached by sequences of maps that consist of a constant map on most of the domain, with a "bubble" (a rescaled harmonic map) concentrating at a single point $a \in \Sigma$ with infinite scale $\lambda \to \infty$.
• Quantitative Stability: The central question is: If a map $v$ has energy close to the infimum ($E(v) \approx 4\pi$), how close is it to this set of "infinitely concentrated" configurations? Specifically, can the distance be bounded by a power of the energy defect $\delta_v = E(v) - 4\pi$?

2. Methodology

The authors develop a novel dynamic approach that simultaneously deforms the map and the domain topology to bridge the gap between the positive genus surface and the limiting punctured sphere.

A. Construction of a Controlled Metric Family

To handle the topological change, the authors construct a 1-parameter family of metrics $G_b = \{g_{\mu,b} : \mu \geq \mu_*\}$ on $\Sigma$ for a fixed point $b \in \Sigma$.

• Weighted Metrics: They define weighted metrics $\bar{g}_{\mu,b}$ that scale a small ball around $b$ up to a hemisphere while shrinking the rest of the surface, effectively turning $\Sigma$ into a punctured sphere as $\mu \to \infty$.
• Diffeomorphism Pull-back: Crucially, they do not work directly with these conformal weights. Instead, they pull them back by a family of diffeomorphisms $T_{\mu,b}$ to create metrics $g_{\mu,b} = T_{\mu,b}^* \bar{g}_{\mu,b}$.
• Efficiency: This construction ensures that the $L^2$-norm of the time derivative of the metric along the deformation path is small ($\|\partial_\mu g_{\mu,b}\|_{L^2} \sim \mu^{-2}$). This allows the domain to be deformed into a punctured sphere with a finite, controlled "distance" in the metric space, which is essential for quantitative estimates.

B. A New Geometric Flow

The authors introduce a gradient flow that evolves both the map $u$ and the metric $g$ simultaneously:
$$\partial_t u = \tau_g(u), \quad \partial_t g = \frac{1}{2} P_{T_g G_b} (k(u,g))$$

• Map Evolution: The map evolves via the harmonic map heat flow (negative gradient of energy with respect to the map).
• Metric Evolution: The metric evolves in the direction of the projection of the energy stress-tensor onto the tangent space of the constructed metric family $G_b$.
• Goal: The flow starts with an almost minimiser and deforms the domain until it becomes a punctured sphere, while the map converges to a harmonic map on this new domain.

C. Łojasiewicz Estimates

A critical component of the proof is establishing sharp Łojasiewicz-Simon inequalities for maps with small energy defect.

• They prove that for maps close to a "bubble tree" configuration, the energy defect is controlled by the squared norm of the gradient of the restricted energy functional:
  $$|E(u,g) - 4\pi| \leq C \|\nabla E_b(u,g)\|_{L^2}^2$$
• This estimate prevents the formation of new singularities (bubbling) during the flow and ensures that the flow travels a finite distance in the configuration space, bounded by $\delta_v^{1/2}$.

3. Key Contributions

1. Topological Deformation Framework: The paper provides a rigorous method to deform a map from a surface of positive genus to a harmonic map on a punctured sphere while controlling the geometric cost of the deformation. This overcomes the limitation of previous methods that relied on the symmetry group of the sphere (Möbius transformations), which does not exist for higher genus.
2. Sharp Quantitative Stability: The authors derive optimal exponents for the stability estimates. They show that the distance to the set of infinitely concentrated minimisers scales exactly as $\delta_v^{1/2}$ (with a logarithmic correction for the $L^2$ distance to the constant).
3. New Gradient Flow: The introduction of a flow that couples the harmonic map heat flow with a specific metric deformation allows for the analysis of problems where the domain topology changes in the limit.

4. Main Results (Theorem 1.1)

For any degree 1 map $v \in H^1(\Sigma, S^2)$ with small energy defect $\delta_v$, there exist a rotation $R \in SO(3)$, a concentration point $a \in \Sigma$, a constant $p \in S^2$, and a scale $\lambda > 1$ such that:

• Concentration Scale: The scale $\lambda$ satisfies $\lambda^{-1} \leq C \delta_v^{1/2}$.
• Bubble Region Approximation: In conformal coordinates around $a$, the map $v$ is close to the rescaled rotation $R\pi_\lambda$:
  $$\|v_a - R\pi_\lambda\|_{H^1(D_\iota, g_{\pi,\lambda})} \leq C \delta_v^{1/2}$$
• Constant Region Approximation: Away from the concentration point, the map is close to the constant $p$:
  $$\|\nabla v\|_{L^2(\Sigma \setminus B_R(a))} \leq C R^{-1} \delta_v^{1/2}$$
• Global $L^2$ Distance: The distance to the constant $p$ satisfies:
  $$\|v - p\|_{L^2(\Sigma)} \leq C \delta_v^{1/2} (1 + |\log \delta_v|)^{1/2}$$

The exponents $1/2$ are proven to be sharp. The logarithmic term in the $L^2$ estimate is also shown to be necessary.

5. Significance

• Resolution of a Singular Limit: This work addresses a fundamental issue in geometric analysis where the set of minimisers is non-compact and the limit involves a change in topology. It provides the first sharp quantitative stability result for such "infinitely concentrated" minimisers.
• Applications: The Dirichlet energy (often called exchange energy) is a key term in models of material science (e.g., liquid crystals, ferromagnets). Understanding the stability of near-minimisers is crucial for predicting the behavior of physical systems where topological defects (bubbles) form.
• Methodological Advancement: The technique of deforming the domain metric to match the topology of the limiting configuration offers a powerful new tool for studying variational problems on manifolds with non-trivial topology, potentially applicable to other problems involving bubbling or concentration phenomena.
• Contrast with Sphere Case: The paper highlights the distinct difficulty of the positive genus case compared to the sphere case ($S^2$), where stability is well-understood due to the rich symmetry group. The authors demonstrate that without these symmetries, a dynamic, metric-deforming approach is required.