Construction of fillings with prescribed Gaussian image and applications

TL;DR

The paper advances the theory of anisotropic energies by providing an explicit construction of fillings with prescribed Gaussian images on the Grassmannian, yielding fillings that are cycles, boundary fillings, or Lipschitz multigraphs under barycenter and orientation conditions. It proves a tight equivalence between polyconvexity and ellipticity for Lipschitz multigraphs, linking polyconvexity to quasiconvexity of associated $Q$-integrands, and demonstrates the necessity of strict polyconvexity for the atomic condition to hold. The construction underpins a broader program to understand regularity and lower semicontinuity properties of anisotropic energies across different classes of competitors, with implications for varifold rectifiability and the porous structure of tangent-plane distributions. Collectively, these results illuminate when elliptic energies admit well-behaved minimizers and how the atomic condition governs convexity-type hypotheses in higher codimension. Mathematical constructs such as the weighted Gaussian image, Lipschitz $Q$-valued multigraphs, and the interaction between currents, varifolds, and polyhedral tilings are central to these insights.

Abstract

We construct $d$-dimensional polyhedral chains such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the barycenter of the prescribed measure, considered as a measure on $\bigwedge^d \mathbb{R}^n$, is $0$) or their boundary is the boundary of a unit $d$-cube (if the barycenter of the prescribed measure is a simple $d$-vector). Such fillings were first proved to exist by Burago and Ivanov [Geom. funct. anal., 2004]; our work gives an explicit construction, which is also flexible to generalizations. For instance, in the case that the measure on the Grassmannian is supported on the set of positively oriented $d$-planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove the surprising fact that, for anisotropic integrands, polyconvexity is equivalent to quasiconvexity of the associated $Q$-integrands (that is, ellipticity for Lipschitz multigraphs) and to show that strict polyconvexity is necessary for the atomic condition to hold.

Figures (3)

• Figure 1: (left) An example with $d=1$, $n=2$. The polyhedral chain $\tilde{S}$ is a sum of sets of parallel $d$--planes such that the corresponding $d$--vectors (in this case, $m_i \omega_{P_i} = (1,1), (1,-1)$, and $(-2,0)$) sum to zero. This condition implies that $\tilde{S}=\partial \tilde{Q}$ for some $(d+1)$--chain $\tilde{Q}$ (in gray). The dotted square is a fundamental domain $F$ for the action of $\mathbb{Z}^n$. (right) A tile $S_F = \partial(\tilde{Q}\mathbin{} F)$. $S_F$ is a $1$--cycle that can be written as the sum of $\tilde{S}\mathbin{} F$. It consists of $d$--planes parallel to the $P_i$'s (thin lines), and $\tilde{Q}\cap \partial [F]$, which is supported on $\partial F$ (thick lines).
• Figure 2: (top) We construct closed surfaces with prescribed tangents by arranging the tiles in a grid. The boundary components of adjacent tiles cancel out, leaving the interior parts, which are parallel to $P_0=\langle (-2,0)\rangle$, $P_1=\langle (1,1)\rangle$, and $P_2=\langle (1,-1)\rangle$. When the grid is large, the interior has much larger mass than the boundary. (bottom) We produce a surface with boundary on $P_0$ and Gaussian image concentrated on $P_1$ and $P_2$ by adding lines parallel to $-P_0$. This cancels out the faces that are parallel to $P_0$. We then add annuli (not shown) so that the boundary of the surface lies on $P_0$. When the grid has large height and even larger width, all but a small fraction of the surface is parallel to $P_1$ or $P_2$.
• Figure 3: We construct a multigraph with prescribed tangents by changing the construction in Figure \ref{['fig:tile-grid']} so that $F$ is a parallelogram. This makes every face of $S_F$ either positively or negatively oriented. To make this a multigraph, we add positively-oriented graphs that cancel out the negatively-oriented faces. These come in two types: graphs parallel to $-P_0$ and graphs containing the negatively-oriented faces on the boundary of the tiling. All of the faces of the resulting surface are positively oriented, so it is a multigraph. We can add annuli (not shown) so that the boundary lies on $P_0$. When the grid has large height and even larger width, most of the lines in the multigraph are parallel to $P_1$ and $P_2$.

Theorems & Definitions (39)

• Theorem 1.1: BurIva
• Theorem 1.2: BurIva
• Definition 1.3
• Remark 1.4
• Theorem 1.5
• Definition 1.6
• Definition 1.7
• Theorem 1.8
• Definition 1.9: DeLellisFocardiSpadaro
• Remark 1.10