## Research Topics

My research focuses on regularity aspects of solutions to geometric variational problems, with particular focus on two classes of problems: Minimal Surfaces and Free-Boundary problems. Recently I have become interested in quantitative inequalities.## Minimal Surfaces

- J. Hirsch, L. Spolaor Interior regularity for two-dimensional stationary Q-valued maps, submitted (2022)
- C. De Lellis, J. Hirsch, A. Marchese, L. Spolaor, S. Stuvard Fine structure of the singular set of area minimizing hypersurfaces modulo p, submitted (2022)
- C. De Lellis, J. Hirsch, A. Marchese, L. Spolaor, S. Stuvard Area minimizing hypersurfaces modulo p: a geometric free-boundary problem, submitted (2021)
- O. Chodosh, Y. Liokumovich, L. Spolaor Singular behavior and generic regularity of min-max minimal hypersurfaces, Ars Inveniendi Analytica (2022)
- N. Edelen, L. Spolaor Regularity of minimal surfaces near quadratic cones, submitted (2019)
- M. Engelstein, L. Spolaor, B. Velichkov (Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currents, Geom. & Topol. (2019)
- M. Colombo, N. Edelen, L. Spolaor The singular set of minimal surfaces near polyhedral cones , JDG (2021)
- M. Colombo, L. Spolaor Quantitative estimate on singularities in isoperimetric clusters, CAG (2019)
- F. Ghiraldin, L. Spolaor On the number of singular points for planar multivalued harmonic functions, Man. Math (2016)
- L. Spolaor Almgren's type regularity for Semicalibrated Currents, Adv. in Math. (2018)
- C. De Lellis, E. Spadaro, L. Spolaor Regularity theory for 2-dimensional almost minimal currents III: blowup, JDG (2019)
- C. De Lellis, E. Spadaro, L. Spolaor Regularity theory for 2-dimensional almost minimal currents II: branched center manifold, Ann. of PDEs (2017)
- C. De Lellis, E. Spadaro, L. Spolaor Regularity theory for 2-dimensional almost minimal currents I: Lipschitz approximation , Transaction of AMS (2016)
- C. De Lellis, E. Spadaro, L. Spolaor Uniqueness of tangent cones for 2-dimensional almost minimizing currents , CPAM (2017)

## Free-Boundary problems

- N. Edelen, L. Spolaor, B. Velichkov A strong maximum principle for minimizers of the one-phase Bernoulli problem, IUMJ (2022)
- G. De Philippis, L. Spolaor, B. Velichkov (Quasi-)conformal methods in two-dimensional free boundary problems, submitted (2021)
- G. De Philippis, M. Engelstein, L. Spolaor, B. Velichkov Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries, submitted (2021)
- L. Spolaor, B. Velichkov On the logarithmic epiperimetric inequality for the obstacle problem, Math. Eng. (2020)
- G. De Philippis, L. Spolaor, B. Velichkov Regularity of the free boundary for the two-phase Bernoulli problem, Inventiones Math. (2021)
- M. Colombo, L. Spolaor, B. Velichkov Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem, Interfaces and Free Boundary (2021)
- L. Spolaor, B. Trey, B. Velichkov Free boundary regularity for a multiphase shape optimization problem, CPDE (2019)
- M. Colombo, L. Spolaor, B. Velichkov On the asymptotic behavior of the solutions to parabolic variational inequalities, CRELLE (2019)
- M. Engelstein, L. Spolaor, B. Velichkov Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional, Duke (2019)
- M. Colombo, L. Spolaor, B. Velichkov Direct epiperimetric inequalities for the thin obstacle problem and applications, CPAM (2019)
- M. Colombo, L. Spolaor, B. Velichkov A logarithmic epiperimetric inequality for the obstacle problem, GAFA (2018)
- L. Spolaor, B. Velichkov An epiperimetric inequality for the regularity of some free boundary problems: the 2-dimensional case, CPAM (2018)

## Quantitative inequalities

- O. Chodosh, M. Engelstein, L. Spolaor The Riemannian Quantitative Isoperimetric Inequality, JEMS (2021)
- M. Engelstein, R. Neumayer, L. Spolaor Quantitative Stability for Minimizing Yamabe Metrics, Transaction of AMS (2021)