The distance between homotopy classes of Sobolev maps on spheres

Technical Summary: The Distance Between Homotopy Classes of Sobolev Maps on Spheres

Problem Statement
This paper addresses the quantitative relationship between distinct homotopy classes of self-maps of the unit sphere $S^n$ ($n \ge 2$) within the critical Sobolev space $W^{1,n}(S^n; S^n)$. Specifically, the authors investigate the "directed distance" between two classes of maps, $E^{(n)}_{d_1}$ and $E^{(n)}_{d_2}$, characterized by their Brouwer degrees $d_1$ and $d_2$. The distance is defined as the supremum over maps $f$ of degree $d_1$ of the infimum over maps $g$ of degree $d_2$ of the $L^n$-norm of the difference in their weak tangential derivatives:
$$\text{Dist}_n(E^{(n)}_{d_1}, E^{(n)}_{d_2}) = \sup_{f \in E^{(n)}_{d_1}} \inf_{g \in E^{(n)}_{d_2}} \left( \int_{S^n} |D_\tau f - D_\tau g|^n \, d\sigma \right)^{1/n}.$$
Prior to this work, while upper bounds and lower bounds with constants depending on specific degrees were known (notably by Brezis, Mironescu, and Shafrir), the exact sharp constant for the general case and the symmetry of this distance for arbitrary degree pairs remained open. In particular, the inequality $\ge$ in the general case was an open problem (Problem 5.9 in Brezis's list).

Methodology
The proof of the main result is divided into establishing sharp upper and lower bounds for the energy difference.

1. Upper Bound (Section 3):
   The authors prove that the distance is bounded above by $\kappa_n |d_1 - d_2|$, where $\kappa_n = n^{n/2} |S^n|$. The strategy involves a "bubble insertion" technique. For a smooth map $f$ of degree $d_1$, the authors construct a map $g$ of degree $d_2$ by locally modifying $f$ on a small ball. This modification replaces the local behavior of $f$ with a "degree bubble" (a map with relative degree $d_2 - d_1$ and minimal energy). The minimal energy required to create a unit of Brouwer degree in the critical space is identified as $\kappa_n$. By utilizing the conformal invariance of the $n$-energy and local flattening lemmas, they demonstrate that the cost to change the degree is exactly proportional to the difference in degrees.

2. Lower Bound (Sections 4–6):
   Proving the reverse inequality requires a more sophisticated approach involving "oscillatory pinning" and "atomic bubbling."

   • Oscillatory Pinning (Section 5): The authors construct a sequence of maps $F_j$ of a fixed degree $d_1$ that are "hard to approximate." This is achieved by rapidly alternating a smooth background map $f_0$ through a finite family of small rotations $Q_\theta$ whose orbits affinely span the ambient space $\mathbb{R}^{n+1}$. If a sequence of maps $g_j$ of degree $d_2$ attempts to approximate $F_j$ with bounded energy, the rapid oscillations force the difference $F_j - g_j$ to converge strongly to zero in $L^n$.
   • Radial Normalization and Bubbling (Section 6): To analyze the energy, the authors define a normalized sequence $u_j$ derived from the difference $F_j - g_j$. While $u_j$ converges to $f_0$ (degree $d_1$) in $L^n$, each $u_j$ retains the degree $d_2$ of $g_j$.
   • Atomic Energy Comparison: The authors invoke a specialized version of the bubbling theorem by Detaille and Van Schaftingen. This theorem states that if a sequence of maps with degree $d_2$ converges to a smooth map of degree $d_1$, the limiting energy measure must contain "atoms" (singularities) carrying at least $\kappa_n |d_1 - d_2|$ of energy. By comparing the atomic part of the limiting energy of the difference $|D(F_j - g_j)|^n$ against the topological defect, they establish that the energy cost cannot be lower than the theoretical minimum.

Key Contributions and Results
The central result is Theorem 1.1, which provides an explicit formula for the directed distance:
$$\sup_{f \in E^{(n)}_{d_1}} \inf_{g \in E^{(n)}_{d_2}} \int_{S^n} |D_\tau f - D_\tau g|^n \, d\sigma = \kappa_n |d_1 - d_2|.$$
Key implications include:

• Symmetry: The result implies that the directed distance is symmetric for all $n \ge 3$, i.e., $\text{Dist}_n(E^{(n)}_{d_1}, E^{(n)}_{d_2}) = \text{Dist}_n(E^{(n)}_{d_2}, E^{(n)}_{d_1})$. This resolves Open Problem 1 from Brezis, Mironescu, and Shafrir (2016) for the case $s=1, p=n$.
• Sharp Constant: The paper identifies the exact constant $\kappa_n$ for the lower bound in the general case, resolving Open Problem 2 from the same source.
• Connection to Heterotopic Energy: The work establishes a formal connection between the "directed distance" functionals and the "heterotopic energy" recently studied by Detaille and Van Schaftingen, showing that the topological defect in the limit of energy measures corresponds precisely to the distance between homotopy classes.

Significance
The paper claims to resolve a long-standing open problem regarding the geometry of homotopy classes in critical Sobolev spaces. By providing an exact, all-degree formula, it extends previous results which were limited to specific dimensions ($n=1, 2$) or specific degree configurations (e.g., opposite signs). The methodology introduces a novel "oscillatory pinning" mechanism to force compactness in the difference of maps, allowing for a uniform comparison of topological costs across all dimensions. The authors note that for $n=2$, this result specifically answers Problem 5.9 from Brezis's list of open problems. The work does not propose new applications or experimental proposals but focuses on the rigorous mathematical characterization of the metric structure of these function spaces.