A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions

Technical Summary: A Smooth Isotopy of Volume-Preserving Diffeomorphisms on Unit Cube Saving Energy Through Extra Dimensions

Problem Statement
The paper addresses the "Two-Point Problem" and the "Classical Variational Problem" within the group of volume- and orientation-preserving diffeomorphisms, $SDiff(M)$, on a Riemannian manifold-with-boundary $(M, g)$. Specifically, it investigates whether, for two configurations $h_0, h_1 \in SDiff(M)$, there exists a smooth isotopy $\{\xi_t\}_{t \in [0,1]}$ connecting them that minimizes the total kinetic energy (action) $J(\{\xi_t\}) = \frac{1}{2} \int_0^1 \|\partial_t \xi_t\|_{L^2}^2 dt$.

While Shnirelman established that for $n \ge 3$, every diffeomorphism on the unit cube $I^n$ is "attainable" (i.e., connected to the identity by some finite-energy path), the existence of an energy-minimizing path is not guaranteed. The paper seeks to construct a specific scenario where a smooth isotopy $\{\xi_t\}$ has infinite total kinetic energy, yet the endpoint $\xi_1$ is attainable via a different smooth isotopy $\{\eta_t\}$ with finite total kinetic energy. This highlights a discrepancy between the energy cost of a specific "wild" construction and the existence of a more efficient path in the ambient space.

Methodology
The author constructs the infinite-energy isotopy $\{\xi_t\}$ by gluing together countably many disjoint "building blocks" supported on arbitrarily small tubular neighborhoods within the unit cube $I^n$ ($n \ge 3$). The methodology relies on three main components:

1. Topological Constraints on Submanifolds: The construction utilizes a compact Riemannian manifold-with-boundary $(M, g)$ with dimension $m \ge 3$, nontrivial first real homology ($H_1(M; \mathbb{R}) \neq \{0\}$), and a fundamental group with a trivial center ($Z(\pi_1(M)) = \{0\}$). Examples include a genus-2 handlebody.
2. Mass Flow Homomorphism: The paper employs the mass flow homomorphism $\theta: Homeo_0(M, \mu) \to H_1(M; \mathbb{R})/\Gamma$ (defined by Fathi). Because the fundamental group has a trivial center, the kernel $\Gamma$ is trivial, making $\theta$ an epimorphism. This allows the construction of isotopies on $M$ that traverse a specific homology cycle $[\sigma]$ an arbitrarily large number of times $N(j)$, thereby forcing the action to grow quadratically with $N(j)$.
3. Ambient Extension and Scaling:
   • Isometric embeddings $\iota: M \hookrightarrow \mathbb{E}^n$ are scaled and translated into disjoint balls $B_j \subset I^n$.
   • Isotopies on $M$ are extended to the ambient space $\mathbb{E}^n$ (and subsequently restricted to $I^n$) using divergence-free vector fields supported on thin tubular neighborhoods of the embedded images.
   • The construction involves a sequence of scales where the number of "turns" $N(j)$ increases sufficiently fast relative to the shrinking size of the support (radius $\rho_j$ and tubular width $\delta_j$).

Key Contributions and Results
The primary result is Theorem 1.3, which states:

• There exists a smooth isotopy $\{\xi_t\}_{t \in [0,1]} \subset SDiff(I^n)$ ($n \ge 3$) such that the total kinetic energy $J(\{\xi_t\}) = \infty$.
• This isotopy is supported on countably many disjoint tubular neighborhoods of homothetic copies of $M$.
• Crucially, the isotopy has "no self-cancellation"; the vector field never stops, and the fluid domain never returns to a previous configuration within the support.
• Despite the infinite energy of $\{\xi_t\}$, the endpoint $\xi_1$ is an attainable diffeomorphism. By Shnirelman's theorem on attainable diffeomorphisms in dimensions $n \ge 3$, there exists another smooth isotopy $\{\eta_t\}$ with $\eta_0 = \xi_0 = id$, $\eta_1 = \xi_1$, and $J(\{\eta_t\}) < \infty$.

The paper demonstrates that the "wild" isotopy $\{\xi_t\}$ is constructed by forcing the fluid to "stir" the low-dimensional submanifold $M$ repeatedly. While this stirring cannot be deformed to reduce energy within the group $SDiff(M)$, the extension to the ambient space $I^n$ (utilizing the extra $n-m$ dimensions) allows for a "shortcut" that bypasses the high-energy cost of the submanifold constraints.

Significance and Claims
The paper claims to provide an explicit example where the "putative fluid flows which minimize kinetic energy" may require non-smooth or measure-theoretic solutions (generalized flows) because the smooth minimizer does not exist for the specific path constructed.

The significance lies in the mechanism of energy saving:

• The infinite energy arises from the topological complexity of $M$ preventing energy reduction within $M$.
• The finite energy alternative exists because the ambient space $I^n$ offers extra dimensions that allow the isotopy to "escape" the constraints of the submanifold, effectively saving energy through the "lift" from the submanifold to the ambient space.

The author notes that this construction is essentially motivated by Shnirelman's work on unattainable diffeomorphisms in 2D and the mass flow homomorphism techniques of Fathi and Eliashberg–Ratiu. The paper also corrects a flawed theorem from a previous version of the work regarding "rigid isotopy extension properties," clarifying that such properties cannot hold because one can always construct an ambient isotopy with arbitrarily small extra energy cost by utilizing thin tubular neighborhoods.

Conclusion
The paper establishes that in dimensions $n \ge 3$, one can construct smooth volume-preserving isotopies with infinite kinetic energy that are nonetheless connectable to the identity by finite-energy paths. This underscores the non-convexity of the energy landscape in $SDiff(M)$ and the role of ambient dimensions in facilitating energy-efficient deformations that are impossible within lower-dimensional submanifolds.