Research
Regularity of higher codimension area-minimizing currents
This project concerns the investigation of the singularities of area-minimizing surfaces in higher codimension. Since the time of Lagrange, the problem of studying the surfaces of least area spanning a given contour has attracted much attention. Apart from the many theoretical applications in the natural sciences (started with the soap films investigations by Plateau), the interest for such geometric objects are closely related to the understanding of many other regularity issues in elliptic partial differential equations, geometric analysis and calculus of variations.
The main contribution to the understanding of higher codimension minimal surfaces is the work by F. Almgren [3]. As it is known since H. Federer's studies [21], area-minimizing surfaces are not regular. In his monumental paper, Almgren proves a celebrated partial regularity result, asserting that the singular set of any area-minimizing current in a Riemannian manifold has Hausdorff codimension at most 2 and is countable for two-dimensional currents (this last assertion was later improved by S. Chang [5] to isolated singularities). Due to its size and complexity, this work, which is considered one of the cornerstones of the entire regularity theory, has attracted much attention in the last years also for its insight into several geometric problems (see, e.g., the case of special Lagrangians and holomorphic subvarieties [4,30,32,31,33,34,39]).
Aim of the project is (1) to provide a new, much shorter proof of this fundamental result: the first steps are given in [14,13,12,37] and the conclusion of the proof is in preparation [15,17,18,19,16]; (2) to develop further the regularity study of the singular set of minimal currents and multiple valued functions (see, e.g., [11]).
Questions in geometric analysis
Isometric immersions
There are many notions of isometry in the low regularity setting. One of the weakest is asking the gradient to be a rotation at almost all the points. Maps with this property have been studied in the context of differential inclusions in geometry and continuum mechanics (see the works by Gromov [22], Eliashberg [20], Müller and Šverák [26], Dacorogna and Marcellini [10]).
However, this notion is not geometrically well suited, because it could collapse entire manifolds to a points. A better definition is to impose that the length of every curve is preserve under the isometry. This kind of notion is not yet well understood (in fact there are different possible generalizations). We are aimed to the investigation of these notions (partial contributions are given in [29] and in collaboration with B. Kirchheim and L.~Székelyhidi [23]) and for other differential inclusions (starting from those coming from calibrated geometries).
Barriers and mean curvature flows with obstacles
It is known that mean-convex sets are local barriers for minimal hypersurfaces for a well-known maximum principle. Moreover, as shown by Meeks and Yau [25], a mean-convex set can also be used for the construction of minimal hypersurfaces. We are interested in the description of all global barriers for minimal surfaces with boundaries on a fixed set. A first result is given in [38] through the asymptotic analysis of a geometric flow, namely a mean curvature flow with obstacle (see also [2]) Part of the project is the development of a theory for such evolution, in terms of weak notions and strong parameterizations, trying to develop a regularity theory in the spirit of mean-convex flows (see, e.g., [40]) and to investigate finer free boundary issues.
Applications in the calculus of variations
Geometric Measure Theory has an important counterpart in the applications, in particular in the context of continuum mechanics, phase transitions, pattern formations etc ...
Following previous works in this directions [9, 35, 36], we are interested in the understanding of several models conjectured to lead to energy driven pattern formation. In particular, a functional we have consider as prototype of such phenomenon is the Ohta--Kawasaki free energy of diblock copolymers. Following the study of Alberti, Choksi and Otto on a sharp interface limit of this functional [1], the uniform distribution for the order parameter and the energy of minimizers was proved in [36]. We are interested in the analysis of specific pattern observed in experiments and simulation, such as the droplets distribution. In [8] we have, indeed, proved a first stability result for a nonlocal isoperimetric problem and we plan to proceed in this direction (see also [6, 7, 24, 28, 27]).
References
Giovanni Alberti, Rustum Choksi, and Felix Otto. Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Amer. Math. Soc., 22(2): 569-605, 2009.
Luis Almeida, Antonin Chanbolle, and Matteo Novaga. Preprint, 2011.
Frederick J. Almgren, Jr. Almgren's big regularity paper, volume 1 of World Scientific Monograph Series in Mathematics. World Scientific Publishing Co. Inc., River Edge, NJ, 2000. Q-valued functions minimizing Dirichlet's integral and the regularity of area-minimizing rectifiable currents up to codimension 2, With a preface by Jean E. Taylor and Vladimir Scheffer.
Costante Bellettini and Tristan Rivière. The regularity of Special Legendrian integral cycles. Ann. Sc. Normale Sup. (to appear), 2012.
Sheldon Xu-Dong Chang. Two-dimensional area minimizing integral currents are classical minimal surfaces. J. Amer. Math. Soc., 1(4):699-778, 1988.
Rustum Choksi and Mark A. Peletier. Small volume fraction limit of the diblock copolymer problem: I. Sharp-interface functional. SIAM J. Math. Anal., 42(3): 1334-1370, 2010.
Rustum Choksi and Mark A. Peletier. Small volume-fraction limit of the diblock copolymer problem: II. Diffuse-interface functional. SIAM J. Math. Anal., 43(2):739-763, 2011.
Marco Cicalese and Emanuele Spadaro. Droplet minimizers to an isoperimetric problem with long-range interactions. Preprint, 2012.
Marco Cicalese, Emanuele Spadaro, and Caterina Ida Zeppieri. Asymptotic analysis of a second-order singular perturbation model for phase transitions. Calc. Var. Partial Differential Equations, 41(1-2): 127-150, 2011.
Bernard Dacorogna and Paolo Marcellini. Implicit partial differential equations. Progress in Nonlinear Differential Equations and their Applications, 37. Birkhäuser Boston Inc., Boston, MA, 1999.
Camillo De Lellis, Matteo Focardi, and Emanuele Spadaro. Lower semicontinuity functionals for almgren's multiple valued functions.Ann. Acad. Scient. Fenn. Math., 36:393-410, 2011.
Camillo De Lellis and Emanuele Spadaro. Higher integrability and approximation of minimal currents. Preprint, 2009.
Camillo De Lellis and Emanuele Spadaro. Center manifold: a study case. Disc. Cont. Din. Syst. A, 31(4):1249-1272, 2011.
Camillo De Lellis and Emanuele Spadaro. Q-valued functions revisited. Mem. Amer. Math. Soc., 211(991):vi+79, 2011.
Camillo De Lellis and Emanuele Spadaro. Multiple valued functions and integral currents. In preparation, 2012.
Camillo De Lellis and Emanuele Spadaro. Optimal regularity for two-dimensional areaminimizing surfaces. In preparation, 2012.
Camillo De Lellis and Emanuele Spadaro. Regularity of area-minimizing currents I: gradient Lp estimates. In preparation, 2012.
Camillo De Lellis and Emanuele Spadaro. Regularity of area-minimizing currents II: center manifold. In preparation, 2012.
Camillo De Lellis and Emanuele Spadaro. Regularity of area-minimizing currents III: blow-up. In preparation, 2012.
Y. Eliashberg and N. Mishachev. Introduction to the h-principle, volume 48 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.
Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.
Mikhael Gromov. Partial differential relations, volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1986.
Bernd Kirchheim, Emanuele Spadaro, and László Székelyhidi, Jr. Typical 1-Lipschitz maps are isometries. In preparation, 2012.
H. Knüpfer and C.B. Muratov. On a isoperimetric problem with a competing non-local term. i. the planar case. Preprint, 2011.
William W. Meeks, III and Shing Tung Yau. The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z., 179(2):151-168, 1982.
S. Müller and V. Šverák. Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. (2), 157(3):715-742, 2003.
C. B. Muratov. Theory of domain patterns in systems with long-range interactions of Coulomb type. Phys. Rev. E (3), 66(6):066108, 25, 2002.
Cyrill B. Muratov. Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions. Comm. Math. Phys., 299(1):45-87, 2010.
A. Petrunin. Intrinsic isometries in Euclidean space. Algebra i Analiz, 22(5):140-153, 2010.
David Pumberger and Tristan Rivière. Uniqueness of tangent cones for semicalibrated integral 2-cycles. Duke Math. J., 152(3):441-480, 2010.
Tristan Rivière. Approximating J-holomorphic curves by holomorphic ones. Calc. Var. Partial Differential Equations, 21(3):273-285, 2004.
Tristan Rivière. A lower-epiperimetric inequality for area-minimizing surfaces. Comm. Pure Appl. Math., 57(12):1673-1685, 2004.
Tristan Rivière and Gang Tian. The singular set of J-holomorphic maps into projective algebraic varieties. J. Reine Angew. Math., 570:47-87, 2004.
Tristan Rivière and Gang Tian. The singular set of 1-1 integral currents. Ann. of Math. (2), 169(3):741-794, 2009.
Emanuele Spadaro. Non-uniqueness of minimizers for strictly polyconvex functionals. Arch. Ration. Mech. Anal., 193(3):659-678, 2009.
Emanuele Spadaro. Uniform energy and density distribution: diblock copolymers' functional. Interfaces Free Bound., 11(3):447-474, 2009.
Emanuele Spadaro. Complex varieties and higher integrability of Dir-minimizing Q-valued functions. Manuscripta Math., 132(3-4):415-429, 2010.
Emanuele Spadaro. Mean-convex sets and minimal barriers. Preprint, 2012.
Clifford Henry Taubes. SW --> Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves [ MR1362874 (97a:57033)]. In Seiberg Witten and Gromov invariants for symplectic 4-manifolds, volume 2 of First Int. Press Lect. Ser., pages 1-97. Int. Press, Somerville, MA, 2000.
Brian White. The size of the singular set in mean curvature flow of mean-convex sets. J. Amer. Math. Soc., 13(3):665-695 (electronic), 2000.