# Geometric Analysis

The basic theme of our research is the study of the analytic, geometric and topological properties of mappings under various analytic assumptions. This often leads to the need to understand the geometries of the underlying spaces. We study modern areas, most of which have their origins in geometric function theory. Mapping problems naturally appear whenever one looks for parametrizations, minimizers, or solutions to analytic equations. Besides analysis, our research is directly connected to several different areas of mathematics, such as differential geometry, partial differential equations, geometric topology, and geometric group theory.

## Members

Academy professor Pekka Koskela

Professor Jani Onninen

Professor Kai Rajala

Professor Mikko Salo

Academy research fellow Enrico Le Donne

Academy research fellow Tapio Rajala

Senior Lecturer Heli Tuominen

Senior Lecturer Juha Lehrbäck

Lecturer Päivi Lammi

**Research interests**

### Analysis on metric spaces

The purpose of this modern area is to develop analysis in non-smooth, or even fractal spaces. New applications to problems where smoothness is not present appear all the time. We study weakly differentiable mappings in singular spaces: Sobolev and BV-functions, Lipschitz and quasiconformal mappings. The Poincare and isoperimetric inequalities as well as heat kernel and other analytic estimates are studied as tools to find out the geometric properties of the underlying spaces. The theory of analysis in metric spaces has shown in particular that surprisingly many deep results in analysis only have little dependence on the structure of the underlying space.

### Geometry of domains

Euclidean domains that admit an extension, say for each p-integrable Sobolev function with p-integrable first order derivatives to the entire Euclidean space can be used as a substitute for the entire space in many questions. Such domains also form natural examples for analysis on metric spaces. Sometimes already the validity of a Poincare type inequality is sufficient. We have studied geometric criteria for extendability of Sobolev functions or for the validity of such inequalities. Many challenging problems remain.

Our research on the Hardy inequalities is also related to the geometry of domains. We study in particular the connections between the validity of these important inequalities and the size and geometry of the boundary of the domain.

### Harmonic maps

### Sobolev mappings and non-linear analysis

Minimizers of suitable energy integrals modeling the physical properties of, say, elastic materials, are sense-preserving Sobolev mappings with some analytic properties. These properties are often familiar in quasiconformal analysis, although weaker. We study the properties of such mappings under minimal regularity assumptions, motivated by the connections to both non-linear elasticity theory and quasiconformal analysis. We try to find the minimal assumptions guaranteeing continuity or partial continuity, invertibility, preservance of sets of measure zero, and regularity of the mappings as well as their inverses; properties which have natural physical interpretations. Similarly, we try to find sharp extensions of the basic results in quasiconformal analysis and geometric function theory.

### Quasiconformal analysis

Quasiconformal and quasiregular mappings extend and generalize conformal maps and complex analytic functions, respectively. In particular, the theory developed for them over the past fifty years extends geometric function theory to higher dimensions. Although the theory is already extensive, there are several fundamental problems that remain unanswered, such as the optimal regularity of mappings, the rigidity of higher dimensional mappings compared to the two-dimensional, boundary behavior, as well as the topological properties. More recently, the theory has been extended to cover mappings between singular spaces, with succesful applications to several problems, for instance in geometric group theory and in the classification of metric spaces.

### Publications

**Some recent publications**

- Z. Balogh, P. Koskela, S. Rogovin: Absolute continuity of quasiconformal mappings on curves, Geom. Funct. Anal., 17 (2007), no. 3, 645--664.
- S. Dendrinos, J. Wright: Fourier restrtiction to polynomial curves I: A geometric inequality, Amer. J. Math., 132 (2010), no. 4, 1031--1076.
- J. Heinonen, P. Pankka, K. Rajala: Quasiconformal frames, Arch. Rational Mech. Anal., 196 (2010), no. 3, 839--866.
- S. Hencl, P. Koskela: Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Rational Mech. Anal., 180 (2006), no. 1, 75--95.
- T. Iwaniec, L. V. Kovalev, J. Onninen: The Nitsche conjecture, J. Amer. Math. Soc., 24 (2011), 345--373.
- T. Iwaniec, N.-T. Koh, L. V. Kovalev, J. Onninen: Existence of energy-minimal diffeomorphisms between doubly connected domains, Invent. Math., to appear.
- T. Iwaniec, J. Onninen: Hyperelastic deformations of smallest total energy, Arch. Rational Mech. Anal., 194 (2009), no. 3, 927--986.
- Piotr Hajasz, Pekka Koskela, Heli Tuominen: Sobolev embeddings, extensions and measure density condition, J. Funct. Anal. 254 (2008), no.5, 1217--1234
- R. Korte, J. Lehrbäck, H. Tuominen: The equivalence between pointwise Hardy inequalities and uniform fatness, Math. Ann., to appear.
- P. Koskela, D. Yang, Y. Zhou: A characterization of Hajlasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions. J. Funct. Anal. 258 (2010), no. 8, 2637-2661.
- K. Rajala: Radial limits of quasiregular local homeomorphisms, Amer. J. Math., 130 (2008), no. 1, 269--289.
- K. Wildrick: Quasisymmetric parametrizations of two-dimensional metric planes, Proc. Lond. Math. Soc. (3) 97 (2008), no. 3, 783--812.