Preprints.
M. Karpukhin, R. Petrides, D. Stern, Existence of metrics maximizing the first Laplace eigenvalue on closed surfaces. arXiv:2505.05293
M. Karpukhin, A. D. Noutchegueme, Optimisation of Steklov transmission eigenvalues and minimal surfaces. arXiv:2509.24055
M. Karpukhin, R. Kusner, P. McGrath, D. Stern, Embedded minimal surfaces in S^3 and B^3 via equivariant eigenvalue optimization. arXiv:2402.13121 An excellent overview article by G. Franz is available here. It summarises the results of this paper within the larger contexts and explains ideas of the proofs. For many beautiful pictures of minimal surfaces related to our results, see M. Schulz's Geometric Analysis Gallery.
Published and accepted articles.
M. Karpukhin, Second Variation Formula for Eigenvalue Functionals on Surfaces. To appear in Pure and Appl. Func. Anal. arXiv:2508.10465
M. Karpukhin, P. McGrath, D. Stern, Large topology asymptotics for spectrally extremal minimal surfaces in S^3 and B^3. Calc. of Var. & PDEs 65:119 (2026), 53 pages. arXiv:2502.10225. DOI 10.1007/s00526-026-03288-3.
M. Karpukhin, A. Métras, I. Polterovich, Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces. IMRN 21 (2024), 13758 - 13784. arXiv:2308.07875. DOI 10.1093/imrn/rnae216.
M. Karpukhin, D. Stern, Existence of harmonic maps and eigenvalue optimization in higher dimensions. Inventiones Mathematicae 236 (2024), 713 - 778. arXiv:2207.13635. DOI 10.1007/s00222-024-01247-3.
M. Karpukhin, M. Nahon, I. Polterovich, D. L. Stern, Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces. J. Differential Geom. 129:2 (2025), 415 - 490. arXiv:2106.15043. DOI 10.4310/jdg/1738163208.
M. Karpukhin, D. L. Stern, Min-max harmonic maps and a new characterization of conformal eigenvalues. J. Eur. Math. Soc. 26:11 (2024), 4071 - 4129. arXiv:2004.04086. DOI 10.4171/JEMS/1401.
M. Karpukhin, D. L. Stern, From Steklov to Laplace: free boundary minimal surfaces with many boundary components. Duke Math. J. 173 (2023), 1557 - 1629. arXiv:2109.11029. DOI 10.1215/00127094-2023-0041.
M. Karpukhin, J. Lagacé, I. Polterovich, Weyl's law for the Steklov problem on surfaces with rough boundary. ARMA 247:5 (2023), 1 - 20. arXiv:2204.05294. DOI 10.1007/s00205-023-01912-6.
M. Karpukhin, J. Lagacé, Flexibility of Steklov eigenvalues via boundary homogenisation. Annales mathématiques du Québec, 48 (2024), 175 - 186. arxiv:2207.02816. DOI 10.1007/s40316-022-00207-8.
M. Karpukhin, A. Métras, Laplace and Steklov extremal metrics via n-harmonic maps. J. of Geom. Analysis 32 (2022), article 154. arXiv:2103.15204. DOI 10.1007/s12220-022-00891-6
M. Karpukhin, D. Vinokurov, The first eigenvalue of the Laplacian on orientable surfaces. Math. Zeitschrift 301 (2022), 2733 - 2746. arXiv:2106.00627. DOI 10.1007/s00209-022-03009-4
M. Karpukhin, X. Zhu, Spherical conical metrics and harmonic maps to spheres. Transactions of AMS 375:5 (2022), 3325 - 3350. arXiv:2104.10304. DOI10.1090/tran/8578
A. Girouard, M. Karpukhin, M. Levitin, I. Polterovich, The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript. J. of Spectral Theory 12:1 (2022), 195 - 225. arXiv:2102.06594. DOI 10.4171/JST/399
M. Karpukhin, N. Nadirashvili, A. V. Penskoi, I. Polterovich, Conformally maximal metrics for Laplace eigenvalues on surfaces. Surveys in Differential Geometry 24:1 (2019), 205 - 256. arXiv:2003.02871. DOI 10.4310/SDG.2019.v24.n1.a6
A. Girouard, M. Karpukhin, J. Lagacé, Continuity of eigenvalues and shape optimization for Laplace and Steklov problems. Geometric and Functional Analysis 31 (2021), 513 - 561. arXiv:2004.10784. DOI 10.1007/s00039-021-00573-5
M. Karpukhin, V. Medvedev, On the Friedlander-Nadirashvili invariants of surfaces. Mathematische Annalen 379 (2021), 1767 - 1805. arXiv:1901.09443. DOI 10.1007/s00208-020-02094-2
G. Cox, D. Jakobson, M. Karpukhin, Y. Sire, Conformal invariants from nodal sets II. Manifolds with boundary. J. of Spectral Theory 11 (2021), 387 - 409. arXiv:1905.06136. DOI 10.4171/JST/345
M. Karpukhin, Index of minimal surfaces and isoperimetric eigenvalue inequalities. Inventiones Mathematicae 223 (2021), 335 - 377. arXiv:1905.03174. DOI 10.1007/s00222-020-00992-5
M. Karpukhin, On the Yang-Yau inequality for the first Laplace eigenvalue. Geometric and Functional Analysis 29:6 (2019), 1864 - 1885 arXiv:1902.03473. DOI 10.1007/s00039-019-00518-z
M. Karpukhin, N. Nadirashvili, A. Penskoi and I. Polterovich, An isoperimetric inequality for Laplace eigenvalues on the sphere. Journal of Diff. Geom. 118:2 (2021), 313 - 333 arXiv:1706.05713. DOI 10.4310/jdg/1622743142
D. Cianci, M. Karpukhin, V. Medvedev, On branched minimal immersions of surfaces by first eigenfunctions. Annals of Global Analysis and Geometry 56:4 (2019), 667 - 690. arXiv:1711.05916. DOI 10.1007/s10455-019-09683-8
M. Karpukhin, Steklov problem on differential forms. Canadian J. Math. 71:2 (2019), 417 - 435. arXiv:1705.08951. DOI 10.4153/CJM-2018-028-6
M. Karpukhin, Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds. Electron. Res. Announc. Math. Sci. 24 (2017), 100 - 109. arXiv:1512.09038. DOI 10.3934/era.2017.24.011
M. Karpukhin, Upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces. Int. Math. Res. Notes 20 (2016), 6200 - 6209. arXiv:1503.08493. DOI 10.1093/imrn/rnv345
M. Karpukhin, Spectral properties of a family of minimal tori of revolution in five-dimensional sphere. Canadian Mathematical Bulletin, 58:2 (2015), 285 - 296. arXiv:1301.2483. DOI 10.4153/CMB-2015-006-0
M. Karpukhin, Nonmaximality of extremal metrics on torus and Klein bottle. Sbornik: Mathematics, 204:12 (2013), 1728 - 1744. arXiv:1210.8122. DOI 10.1070/SM2013v204n12ABEH004358.
M. Karpukhin, G. Kokarev, I. Polterovich, Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces. Annales de l'Institute Fourier, 64:6 (2014), 2481 - 2502. arXiv:1209.4869. DOI 10.5802/aif.2918.
M. Karpukhin, Spectral properties of bipolar surfaces to Otsuki tori. J. of Spectral Theory, 4:1 (2014), 87 - 111. arXiv:1205.6316. DOI 10.4171/JST/63.