# Quasiconformal Homogeneity: Energy Methods and Sharp Bounds (07frg127)

Arriving in Banff, Alberta Sunday, March 4 and departing Sunday March 11, 2007

## Organizers

Petra Bonfert-Taylor (Wesleyan University)

Edward Taylor (Wesleyan University)

## Objectives

We will provide both background and descriptions of the two programs briefly detailed above.

An (orientable) hyperbolic manifold $M$ is $K$-quasiconformally homogeneous if, given any pair of points $x$ and $y$ in $M$, there exists a $K$-quasiconformal homeomorphism $f: M rightarrow M$ such that $y = f(x)$. If there exists a $K geq 1$ so that $M$ is $K$-quasiconformally homogeneous, then $M$ is said to be uniformly quasiconformally homogeneous. The study of uniformly quasiconformally homogeneous planar domains was initiated by Gehring and Palka in [6]. This study was extended to the study of uniformly quasiconformally homogeneous hyperbolic manifolds by Bonfert-Taylor, Canary, Martin and Taylor in [2], and continued in subsequent work by Bonfert-Taylor, Bridgeman, Canary and Taylor ([1].) The work proposed herein builds on these studies, and in particular follows from:

Theorem 1: ([2].) For each dimension $n geq 3$ there exists a constant $K_n > 1$, depending only on the dimension $n$, so that any non-trivial uniformly quasiconformally homogeneous hyperbolic manifold $M$ has the property that $K(M) geq K_n.$

Informally, one thinks of a quasiconformal mapping as a mapping that almost everywhere distorts infinitesimal spheres to infinitesimal ellipsoids of bounded eccentricity. In the statement above $K(M)$ is the least value of $K$ so that $M$ is $K$-quasiconformally homogeneous. The proof of this theorem depends on the phenomenon of quasiconformal rigidity in dimensions $n geq 3$; this phenomenon is well-known to fail in dimension two. A fundamental conjecture in the study of quasiconformal homogeneity, and a focus of this proposal, is:

Conjecture 2: There exists a constant $K_2 > 1$ so that if $S$ is a closed hyperbolic surface then $K(S) geq 2$.

We note that there has been recent progress on verifying this conjecture for large classes of surfaces, e.g. for instance, it is now known that there exists a uniform constant $K_{he}> 1$ so that every hyperelliptic surface $S$ of arbitrary genus has $K(S) geq K_{he}$ ([1].) However the techniques used to make this progress will not, in and of themselves, result in the complete resolution of Conjecture 2. New ideas and techniques are needed, and indeed we believe that techniques and ideas coming out of the harmonic maps and energy methods may well lead to significant progress.

PART 1: Harmonic maps and energy methods

We have shown that mappings between Riemann surfaces which minimize mean distortion have inverses which minimize total energy, that is harmonic mappings:

Theorem 3: Let $Sigma_1$ and $Sigma_2$ be two closed homeomorphic hyperbolic surfaces. If there is a homeomorphism $f:Sigma_1toSigma_2$ with smallest mean distortion, then its inverse is harmonic. Moreover this harmonic mapping has smallest total energy. Conversely, a harmonic homeomorphism of smallest total energy has an inverse of smallest mean distortion.

Note that in general the inverse of a harmonic mapping between Riemann surfaces is not a minimizer of the mean distortion even if it has finit energy. (Compare Trapani - Valli [13].) Now as a corollary of the well-known results of Eells - Sampson [4], Hartman [8], Schoen - Yau [11], Sampson [10] and Jost - Schoen [9], we obtain the existence of a unique harmonic mapping of smallest total energy in any homotopy class of mappings between Riemann surfaces.

Theorem 4: Let $Sigma_1$ and $Sigma_2$ be two homeomorphic closed hyperbolic surfaces. Let $phi:Sigma_1to Sigma_2$ be a diffeomorphism. Then there is a diffeomorphism $f:Sigma_1toSigma_2$ of smallest mean distortion homotopic to $phi$, and its inverse is thus a harmonic mapping of smallest total energy.

We conjecture that there is a certain gap in the energy spectrum of any closed hyperbolic Riemann surface. In particular, we wish to prove:

Conjecture 5: There exists a number $E_0>1$ and a constant $lambdageq 84$ such that any closed hyperbolic Riemann surface $S$ of genus $ggeq 2$ satisfies the following: The number of energy-minimizing harmonic maps of $S$ to itself with total energy bounded above by $4 pi E_0 cdot(g-1)$ is at most $lambdacdot g$.

Weaker versions of the conjecture are immediate for any fixed genus. To see this, consider the real-valued function $E: mathcal{T}_g to mathbb{R}$ of the Teichm"uller space $mathcal{T}_g$, defined by
setting $E(Sigma_2)$ to be the energy of the unique harmonic map between $Sigma_1$ and $Sigma_2$ in some fixed (degree one) homotopy class. The function is proper [15] (see also [5]), and since the mapping class group acts properly discontinuously on $mathcal{T}_g$, we see that but a finite number of energy minimizing maps from $Sigma$ to itself have energy less than a fixed $E_0$.

This observation holds out some hope for the proof of the conjecture, as the properness proofs in [15], [11] and [5] rely in a fundamental way only upon estimates in a single collar. Thus, one might try to refine the estimate to avoid any strong dependence on the genus of the underlying surface $Sigma$. (Indeed, there are already some weak estimates in [7] on the energy density of a harmonic map that do not refer to the genus of the surface $Sigma$, but only to the quasisymmetric norm of the maps induced on the circle of infinity of hyperbolic space by the lift of the harmonic map.)

We observe that a positive resolution to Conjecture 5 would imply the existence of a constant $K_2>1$ such that every $K$-quasiconformally homogeneous closed hyperbolic surface satisfies that $Kgeq K_2$. The idea here is that a quasiconformal mapping with a small (i.e. close to $1$) dilatation $K$ has an inverse with low energy. However the linear upper bound in the genus $g$ in the number of low energy maps, along with the linearity of (hyperbolic) area in the genus $g$ via the Gauss Bonnet theorem, implies, for any given genus $g$, that there are never enough small dilatation maps to find a sequence ${S_i}$ of closed hyperbolic surfaces so that $K(S_i) rightarrow 1$.

An equally interesting question remains as to whether, on the other hand, the existence of a constant $K_2$ as described would imply the conclusion of Conjecture 5.

PART 2: Hyperbolic manifolds of minimal quasiconformal homogeneity

Our goal in Objective (2) is to finish work that (conjecturally) has identified the sharp lower bound $K_3>1$ on the quasiconformal homogeneity constant of non-elementary uniformly quasiconformally
homogeneous hyperbolic $3$-manifolds, as well as the structure of manifolds that come close to achieving this lower bound. In particular, we ask:

Question 6: Which uniformly quasiconformally homogeneous hyperbolic $3$-manifolds $M$ have a homogeneity constant close to $K_3$? For such manifolds $M$, is necessarily true that $K(M) > K_3$?

As a model for this study, one can consider a modification of the definition of quasiconformal homogeneity as applied to orientable hyperbolic surfaces. We say that such a surface $S$ is $K$-quasiconformally homogeneous rel conformal if there exists a constant $K geq 1$ so that for each pair $x,y in S$ there exists a $K$-quasiconformal automorphism $f$ that a) is homotopic to a conformal automorphism of $S$, and b) satisfies $y = f(x).$ We will use the notation $K_{conf}(S)$ for the best'' dilatation over all pairs of points for such a surface.

As a part of the work that we wish to complete during our visit to BIRS, we have recently shown:

Theorem 7 (Bonfert-Taylor, Martin, Reid, and Taylor [3]) Let [ K_c = inf{K_{conf}(S): mbox{S is a closed hyperbolic surface}}, ] then $K_c$ is computed in terms of the diameter of the $(2,3,7)$-triangle orbifold, it is strictly greater than one, and $K_{conf}(S) > K_c$ for each such $S$.

A key point in proving this result is a careful analysis of an extremal result of Teichm"uller [12]. In brief, one shows the emph{strict} inequality $K_{conf}(S) > K_c$ by explicitly computing the line element field associated to Teichm"uller's minimal dilatation map associated to the diameter of the $(2,3,7)$-triangle
group and then observing that this line element field is not invariant with respect to any non-elementary conformal group action. We note that in dimensions three and above, quasiconformal rigidity (see Theorem~ref{Kn>1}) allows us to deal with quasiconformal homogeneity without the restriction rel conformal as we currently must in dimension two. Recall that we defined $K_n = inf {K(M): Mneqmathbb{H}^n textrm{ is a quasiconformally homogeneous$n$-manifold}}$. Following the above theorem, it is natural to conjecture for $K_{n=3}$ that:

Conjecture 8: $K_3$ is computed in terms of the diameter of a $mathbb{Z}_2$-extension of the orientation-preserving subgroup of the $(3,5,3)$ Coxeter reflection group. We further conjecture that
$K(M)$ is strictly greater than $K_3$ for any uniformly quasiconformally homogeneous hyperbolic $3$-manifold $Mneqmathbb{H}^3$.

Our first step in verifying this conjecture is to show that the $mathbb{Z}_2$-extension of the orientation-preserving subgroup of the $(3,5,3)$-Coxeter reflection group has the minimal diameter
amongst all closed hyperbolic $3$-orbifolds. Having verified this one can then use quasiconformal rigidity and analytic arguments to show that the least uniform quasiconformal homogeneity constant is a computable value- this is $K_3$- of this diameter.

We will thus be able to conclude that for any uniformly quasiconformally homogeneous hyperbolic $3$-manifold $M$ that $K(M) geq K_3$. To verify the full conjecture, we need to establish the structure of certain extremal quasiconformal maps in space. It is here that geometric and topological properties of uniformly quasiconformally homogeneous hyperbolic manifolds has led us to a certain natural questions in geometric function theory. A conjecture we deem achievable (Conjecture 9) is given below, and a more ambitious question (Question 10) is then broached; we will be focussing intently on both during our stay.

The key piece of missing information needed to obtain strict inequality in Conjecture 8 is the appropriate generalization to space of Teichm"uller's work. Here is a description of the extremal problem Teichm"uller solved in dimension $n=2$. Let $rin (0,1)$. Amongst all quasiconformal mappings of the unit disk to itself that extend to the identity on the boundary of the disk and map the
origin to the point $-r$, which has the least dilatation? Teichm"uller explicitly constructed the extremal map and showed its uniqueness in the planar case. The higher dimensional analog of this problem is currently open. A first attempt was made by Anderson and Vamanamurthy in published work dating from the late 1970's. Under the additional assumption that the maps under consideration fix a
plane containing the origin and the points $(x,0,0) : |x|<1$, Anderson and Vamanamurthy assert that emph{the rotation of the $2$-dimensional extremal map is the solution to the extremal problem in dimension $n=3$ with respect to linear dilatation}.

We note that in higher dimensions there are several notions of quasiconformal dilatation that do not agree with each other, namely linear dilatation, outer and inner dilatation and maximal dilatation, whereas in dimension $n=2$ these dilatations all agree with each other. Recall that the texttt{linear dilatation} of a quasiconformal mapping $f:Uto V$ at a point $xin Usubsetmathbb{R}^n$ is given by
[ H(x,f) = limsup_{rto 0} frac{max{|f(y)-f(x)| : |y-x|=r}}{min{|f(y)-f(x)| : |y-x|=r}}. ]
Furthermore, the texttt{inner dilatation} $K_I$ and the texttt{outer dilatation} $K_O$ are given by
K_O(f) = sup frac{M(Gamma)}{M(f(Gamma))}, ]
where the suprema are taken over all path families $Gamma$ in $U$such that the moduli $M(Gamma)$ and $M(f(Gamma))$ are not simultaneously $0$ or $infty$. The texttt{maximal dilatation} $K(f)$ is the dilatation we have been considering for the homogeneity problem of manifolds and is defined as the maximum of outer and inner dilatations. See [14] for details. We note that a homeomorphism $f:Uto V$ between subsets of $overline{mathbb{R}^n}$ is quasiconformal if one (and hence both) of the dilatations $K_I(f)$, $K_O(f)$ are finite, or equivalently, if the linear dilatation $H(x,f)$ is bounded for all $xin U$.

We conjecture:

Conjecture 9: The rotation of Teichm"uller's extremal map is in fact the extremal map in dimension $n=3$ with respect to linear dilatation.

We strongly believe in the validity of this conjecture. However, it would be very interesting if the extremal problem had a different solution with respect to maximal dilatation. On the other hand, if the extremal solution were independent of the dilatation considered, then the extremal map would no longer have constant maximal dilatation, which by itself would also be very different from the corresponding result in two dimensions. It is natural to ask:

Question 10: Does the extremal problem in dimension $n=3$ have the same solution, independently of the dilatation considered? In particular, is the maximal dilatation minimized for the rotation of
Teichm"uller's extremal map?

BIBLIOGRAPHY:

[1] P. Bonfert-Taylor, M. Bridgeman, R. Canary and E. Taylor, "Quasiconformal homogeneity of hyperbolic surfaces with fixed-point full automophisms", submitted.

[2] P. Bonfert-Taylor, R. Canary, G. Martin, and E. Taylor, "Quasiconformal homogeneity of hyperbolic manifolds", Math. Ann. 331 (2005), pp. 281--295.

[3] P. Bonfert-Taylor, G. Martin, A. Reid and E. Taylor, "Hyperbolic manifolds of minimal quasiconformal distortion", preprint.

[4] J. Eells and J.H. Sampson, "Harmonic mappings of Riemannian manifolds", Amer. J. Math. 86 (1964), pp. 109--160.

[5] A. Fischer and A. Tromba, "A new proof that Teichm"uller space is a cell", Trans. Amer. Math. Soc. 303 (1987), pp. 257--262.

[6] F. Gehring and B. Palka, "Quasiconformally homogeneous domains", J. Analyse Math. 30 (1976), pp. 172--199.

[7] R. Hardt and M. Wolf, "Harmonic extensions of quasiconformal maps to hyperbolic space", Indiana U. Math. J. 46 (1997), pp. 155--163.

[8] P. Hartman, "On homotopic harmonic maps", Canad. J. Math. 19 (1967), pp. 673--687.

[9] J. Jost and R. Schoen, "On the existence of harmonic diffeomorphisms", Invent. Math. 66 (1982), no. 2, pp. 353--359.

[10] J.H. Sampson, "Some properties and applications of harmonic mappings", Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, pp. 211--228.

[11] R. Schoen and S. T. Yau, "Existence of incompressible minimal surfaces and the topology of $3$-dimensional manifolds with non-negative scalar curvature", Ann. of Math. 110 (1979), pp. 127--142.

[12] O. Teichm"uller, "Ein Verschiebungssatz der quasikonformen Abbildung", Deutsche Math. 7 (1944), 336--343.

[13] S. Trapani and G. Valli, "One-harmonic maps on Riemann surfaces", Comm. Anal. Geom. 3 (1995), no. 3-4, pp. 645--681.

[14] J. Väisälä, "Lectures on $n$-dimensional quasiconformal mappings." Lecture Notes in Mathematics, Vol. 229. Springer-Verlag, Berlin-New York, 1971.

[15] M. Wolf, "The Teichm"uller theory of harmonic maps", J. Diff. Geometry 29 (1989), pp. 449--479.