Hyperconvex curve

In mathematics , hyperconvex curves are certain curves in projective space that are important in the representation theory of groups of surfaces.

Hyperconvex curves in projective space

Be . The projective space is the space of all 1-dimensional subspaces of the . A closed curve ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mathbb {R} P^{n-1}=P(\mathbb {R} ^{n})}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \gamma :S^{1}\rightarrow P(\mathbb {R} ^{n})}$

is called hyperconvex if for each tuple we have pairwise different points: ${\displaystyle n}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$

${\displaystyle \mathbb {R} ^{n}=\gamma (x_{1})\oplus \ldots \oplus \gamma (x_{n})}$,

in other words: if none is contained in the linear envelope of . ${\displaystyle \gamma (x_{i})}$ ${\displaystyle \gamma (x_{j}),j\not =i}$

Frenet curves

A hyperconvex curve is called a Frenet curve if it is a family of maps ${\displaystyle \gamma :S^{1}\rightarrow P(\mathbb {R} ^{n})}$ ${\displaystyle (\gamma ^{1},\ldots ,\gamma ^{n-1})}$

${\displaystyle \gamma ^{p}:S^{1}\rightarrow Gr(p,n)}$

into the Grassmann manifold such that ${\displaystyle Gr(p,n)}$

• ${\displaystyle \gamma ^{1}=\gamma }$
• for and all tuples of pairwise different points is a direct sum ${\displaystyle l_{1}+\ldots +l_{r}\leq n}$ ${\displaystyle r}$ ${\displaystyle (x_{1},\ldots ,x_{r})}$ ${\displaystyle \gamma ^{l_{1}}(x_{1})+\ldots +\gamma ^{l_{r}}(x_{r})}$
• for and against each sequence converging pairs of r-tuples of different points is . ${\displaystyle l_{1}+\ldots +l_{r}\leq n}$ ${\displaystyle (x,\ldots ,x)}$ ${\displaystyle (X_{i})_{i\in N}}$ ${\displaystyle X_{i}=(x_{1,i},\ldots ,x_{r,i})}$ ${\displaystyle \lim _{i\to \infty }\bigoplus _{s=1}^{r}\gamma ^{l_{s}}(x_{s,i})=\gamma ^{l_{1}+\ldots +l_{r}}(x)}$

Note that the through are determined uniquely. If it is differentiable any number of times, then is the subspace spanned by , so the term corresponds to the term of a Frenet curve , which is commonly used in differential geometry . ${\displaystyle \gamma ^{p}}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma \colon S^{1}\to \mathbb {R} ^{n}}$ ${\displaystyle \gamma ^{p}(x)}$ ${\displaystyle \gamma (x),\gamma ^{\prime }(x),\gamma ^{\prime \prime }(x),\ldots ,\gamma ^{(p-1)}(x)}$

Hitchin component

The Hitchin component is a connected component of the representation variety of a surface group in , see higher Teichmüller theory , which was originally described by Hitchin with the help of Higgs bundles . The Hitchin component becomes accessible to a geometrical investigation by the following theorem of Labourie: ${\displaystyle \pi _{1}S}$ ${\displaystyle PSL(n,\mathbb {R} )}$

If a representation of a group of surfaces belongs to the Hitchin component, then there is a hyperconvex Frenet curve ${\displaystyle \rho :\pi _{1}S\rightarrow PSL(n,\mathbb {R} )}$

${\displaystyle \gamma :S^{1}=\partial _{\infty }\pi _{1}S\rightarrow P(\mathbb {R} ^{n})}$,

which - is equivariant with respect to the canonical effect of on its edge in infinity and of on . One can show that every equivariate hyperconvex curve is a Frenet curve. (Laborie) ${\displaystyle \rho }$ ${\displaystyle \pi _{1}S}$ ${\displaystyle \partial _{\infty }\pi _{1}S=S^{1}}$ ${\displaystyle PSL(n,\mathbb {R} )}$ ${\displaystyle P(\mathbb {R} ^{n})}$

Representations for which an equivariate hyperconvex curve exists are referred to as hyperconvex representations .

The converse also applies: if a representation is hyperconvex, then it belongs to the Hitchin component. (Guichard)

literature

• François Laborie : Anosov flows, surface groups and curves in projective space. Invent. Math. 165 (2006), no. 1, 51-114. pdf
• Olivier Guichard: Composantes de Hitchin et représentations hyperconvexes de groupes de surface. J. Differential Geom. 80 (2008), no.3, 391-431 pdf