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Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that maps $A \mapsto A'$, $B \mapsto B'$, $C \mapsto C'$. What is the energy of $f$?

Now let me write some precisions to make this question intelligible.

Given a point $x \in \mathbb{H}^2$ inside $T$ (the convex hull of $\{A, B, C\}$), this point $x$ has uniquely defined barycentric coordinates $\alpha$, $\beta$, $\gamma$. These are nonnegative real numbers such that $\alpha + \beta + \gamma = 1$ and $x$ is the Riemannian center of mass of the points $A$, $B$, $C$ with weights $\alpha$, $\beta$, $\gamma$ 1.

Okay so now the "barycentric interpolation" $f : T \to T'$ is defined as the map which is "the identity in barycentric coordinates". In other words, $f$ sends a point $x \in T$ with barycentric coordinates $\alpha, \beta, \gamma$ to the point $f(x) \in T'$ such that $\alpha$, $\beta$ and $\gamma$ are the barycentric coordinates of $f(x)$ in the triangle $T'$.

What I would like to have, if possible, is an explicit expression of the energy of this map. In general, the energy of a map between Riemannian manifolds $f : (M,g) \to (N,h)$ is defined as $$ E(f) = \frac{1}{2} \int_M \Vert df \Vert_{g,h} ^2 d\mathrm{vol}_g~. $$

Here $\Vert \cdot \Vert_{g,h} ^2$ denotes the "Hilbert-Schmidt norm"2 of the linear map $df : (T_xM, g) \to (T_{f(x)}N, h)$.

Note that a harmonic map is defined as a critical point of this energy functional. Here I am unsure whether this barycentric interpolation is a (or "the unique" in the appropriate sense) harmonic map from $T$ to $T'$, I haven't thought too hard about that. I did try quite hard to compute the energy of that map, unsuccessfully. But I am not yet convinced that it is impossible to find an explicit expression of this energy. Let me know if you would like me to give details on the computations I did. In my dreams, the energy would have an expression of the form: $$ E(f) = w_1 \, d(A', B')^2 + w_2\, d(B', C')^2 + w_3\, d(A', C')^2 $$ where $w_1$, $w_2$ and $w_3$ are nonnegative real numbers (that possibly sum to $1$) that only depend on the geometry of the domain triangle $T$. More generally, I would be happy with some explicit function in the three numbers $d(A', B')$ $d(B', C')$ and $d(A', C')$, where the function would only depend on the geometry of $T$.


1 By definition, this means that, equivalently:

  • $x$ minimizes the function $x \mapsto \alpha \,d(x,A)^2 + \beta \,d(x,B)^2 + \gamma \,d(x,C)^2$.
  • $x$ satisfies $\alpha \, \overrightarrow{xA} + \beta \, \overrightarrow{xB}+ \gamma \, \overrightarrow{xC} = 0$. Here I use the following (nonstandard) notation: for points $M, P \in \mathbb{H}^2$, I let $\overrightarrow{MP}$ denote the tangent vector in $T_M \mathbb{H}^2$ such that $\exp_M(\overrightarrow{MP}) = P$.

2 This norm can be defined as the "trace with respect to $g$" of the symmetric bilinear form on $T_xM$ defined by $(u,v) \mapsto h(df(u), df(v))$. In local coordinates on $M$ and $N$, the norm is given by: $$\Vert df \Vert^2 = h_{\alpha \beta} \frac{\partial f^\alpha}{\partial x^i} \frac{\partial f^\beta}{\partial x^j} g^{ij} ~.$$

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