Selection of scientific publications

1. A Jordan algebra approach to the eiconal equation,

   We establish a natural correspondence between (the equivalence classes of) cubic solutions of \[\sum_{i,j=1}^k Q^{ij}\,\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}=9(\sum_{i,j=1}^k Q_{ij}x_ix_j)^2\] and (the isomorphy classes of) cubic Jordan algebras.

2. Minimal cubic cones via Clifford algebras, Complex Anal. and Operator Theory, 4(2010) to appear

   In J. Diff. Geom. (1967) W.-Y. Hsiang posed the following problems on minimal algebraic cones: How does one classify irreducible minimal cubic forms in n variables, n ≥ 4, with respect to the natural action of the orthogonal group? A weaker question, whether there are always irreducible minimal cubic forms in n variables for all n ≥ 4.

   We construct two infinite families of algebraic minimal cones in Rⁿ, n ≥ 4. The first family consists of minimal cubics given explicitly in terms of the Clifford algebras. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any n ≥ 4, n ≠ 16k+1, there is at least one minimal cone in Rⁿ given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in R^m2 defined by an irreducible homogeneous polynomial of degree m.

3. Cartan's theorem for general cubic forms, Proc. Amer. Math. Soc., 138(2010), 2889-2895

   Recall that a hypersurface is called isoparametric if it has constant principal curvatures. In 1939, Cartan established a remarkable result that there exist exactly four different isoparametric hypersurfaces with three distinct principal curvatures: their dimensions are equal to 3d, where d=1,2,4,8, they are algebraic and the corresponding defining polynomials f_d can be naturally expressed in terms of the multiplication in one of four classical real division algebras F_d of dimension d, where F₁=R (reals), F₂=C (complexes), F₄=H (quaternions) and F₈=O (octonions). These polynomials are given explicitly by

   f_d=x_n³- ³/₂ x_n(u₀ u₀^*+u₁ u₁^*- 2u₂ u₂^*-2x_n-1 ²)+3√3/2[x_n-1(u₀ u₀^* -u₁ u₁^*)+(u₀u₁)u₂+ u₂^*(u₁^*u₀^*)],

   where x=(u₀,u₁,u₂,x_n-1,x_n), and the numbers u_i⊂ F_d should be interpreted as vectors in R^d. By the Cartan theorem, these polynomials are the only (up to congruence in R^3d+2) cubic solutions of the Cartan-Münzner system

   |∇ f|²=9|x|⁴,     Δ f=0.

   We obtain a generalization of the Cartan theorem by showing that, dropping the harmonicity condition, any cubic polynomial solutions satisfying the first equation |∇ f|²=9|x|⁴ alone is either a reducible cubic

   x³_n-3x_n(x²₁+  ···  x²_n-1)

   or one of the four Cartan isoparametric cubics.

4. Disjoint minimal graphs, Anal. Global Analysis and Geom., 35(2009), no 2, 139-155. available by arXiv

   We prove that the number of disjoint minimal graphs supported on domains in n-dimensional Euclidean space has a polynomial bound as a function of dimension: e(n+1)². In the two-dimensional case we show that the maximal number of disjoint graphs is 3 (a conjecture of W. Meeks states that the optimal bound is 2).

5. On the Jacobian of the harmonic moment map, Complex Anal. and Operator Theory , 3 (2009), no 2, 399-417(with B. Gustafsson), available by arXiv

   In this paper we represent harmonic moments in the language of transfinite functions, that is projective limits of polynomials in infinitely many variables. We obtain also an explicit formula for the Jacobian of a generalized harmonic moment map.

6. The resultant on compact Riemann surfaces, Comm. Math. Physics, 286(2009), 313-358 (with B. Gustafsson) available by arXiv

   The classical resultant of two polynomials is an active tool in elimination theory. We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, we prove coincidence of the exponential transform of a quadrature domain D and the (meromorphic) resultant of two meromorphic functions (on the Schottky double of the domain) canonically associated with D. At the operator theoretic level the meromorphic resultant reads as a Fredholm determinant in an appropriate functional calculus and from an algebraic point of view it may be understood as a `integrated' local (tame) index. These properties make the meromorphic resultant very useful for operator theoretic applications; in particular we demonstrate some relations of the resultant to infinite Toeplitz determinants and give its cohomological interpretation.

7. Ullemar formula for the moment map, II, Linear Algebra and Appl., V. 404(2005), 380-388.  
8. Ullemar's formula for the Jacobian of the complex moment mapping, Complex Variables and Appl., 49(2004), 55-72. (with Kuznetsova O.S. )

   In these two papers we study the harmonic moments of a domain which are the Taylor coefficients of the corresponding Cauchy transform. If a domain is given as an image of the unit disk under a polynomial map then its moments form a finite system which is called the moment map and it sends the polynomial coefficients to the moments. In 1980 C. Ullemar conjectured a closed formula for the Jacobian of the harmonic moment map for polynomials with real coefficients in terms of some Hurwitz determinants. We establish a complete proof of the Ullemar formula as well as some its generalizations.

9. Positive definite collections of disks, Indiana Univ. Math. J. , V. 55(2006), No 6, 380-388.

   In the paper we answer affirmative to one question of B. Gustafsson and M. Putinar concerning the positivity of the defining kernel associated with a finite collection of two-dimensional disks. The main result of the paper gives a sharp estimate in terms of roots of Bessel functions for the size of collection to ensure the positive definiteness of the kernel.

10. Algebraic structure of quasiradial solutions to the p-harmonic equation. Pacific J. Math., 226(2006), No 1, 179-200.

    In 1984 G. Aronsson has constructed a discrete family of quasiradial symmetric solutions to the p-Laplace equation. For a given p>1 there is a whole family of the solutions which is counted by a natural N>1, the so-called N-solutions which can be thought of as non-linear analogues of harmonic polynomials. We obtain an explicit representation for these N-solutions and show that any N-solution is some degrees of an algebraic function in the plane. For certain values of N solutions are purely algebraic and we give a complete description for these N. We show that such algebraic solutions obey some interesting extremal features. In particular, the so-called minimal series of the algebraic solutions admits a nice gas dynamic interpretation in terms of the adiabatic exponents of an ideal gas.

11. Subharmonicity of higher dimensional exponential transforms. Operator Theory: Advances and Appl., Birkhauser, Vol. 156(2005), 257-277.

    With any domain in Rⁿ one can associate a renormalized Riesz potential, the so-called exponential transform. This object has operator theoretic roots and in the two-dimensional case comes back to the well-known principal function introduced by J. Pincus for operators with commutator in trace class. This function plays a key role in the discovered by M. Putinar correspondence between potential and operator theory on quadrature domains. In the higher dimensions however the exponential transform is studied not well. In this paper we give a solution to one conjecture due to B. Gustafsson and M. Putinar about subharmonicity of the higher-dimensional exponential transform. The proof is based on a combination of rearrangement and co-area techniques. A crucial step in the proof is a new non-trivial version of the classical Cauchy inequality to functions with prescribed uniform norms.

12. Denjoy-Ahlfors theorem for harmonic functions on Riemannian manifolds and external structure of minimal surfaces, Comm. Anal. Geom. 4(1996), no. 4, 547-587. (with Miklyukov V.M.)

    The well-known Denjoy-Ahlfors theorem gives an estimate on the number of different asymptotic tracts of a entire holomorphic function in terms of its growth characteristics. Here we give a generalization of this result for subharmonic functions on arbitrary Riemannian manifolds. The main tool is the so-called N-means of the fundamental frequency of an open set on a Riemannian submanifold (the averaged arithmetic means of the fundamental frequency over all decomposition of the set into N components). This concept for higher dimensional Euclidean spaces was introduced and virtuously applied by V. Miklyukov in the late 1970:s for studying of asymptotic topological structure of (sub)solutions to a wide class of quasilinear equations. We extend this method on the Riemannian case and obtain versions of the classical Liouville theorem for p-harmonic functions without requirement of geodesic completeness on a manifold. As a corollary of our method we exhibit several upper estimates on the topological index of the height function on minimal surfaces in Rⁿ. In particular this yields a new prove of the celebrated Bernstein's result on triviality of two-dimensional minimal graphs. Other applications to the theory of minimal surfaces are also discussed.