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Showing papers in "Crelle's Journal in 2020"


Journal ArticleDOI
TL;DR: In this article, an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras was developed, and it was shown that Schellekens' classification of $V_1$-structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operators.
Abstract: We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens' classification of $V_1$-structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds.

67 citations


Journal ArticleDOI
TL;DR: The cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties were studied in this paper.
Abstract: We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac-Moody algebras.

43 citations


Journal ArticleDOI
TL;DR: In this paper, a uniform curvature estimate for stable free boundary minimal hypersurfaces with a uniform area bound is presented, which is a natural generalization of the celebrated Schoen-Simon-Yau interior curvature estimates up to the free boundary.
Abstract: In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces which satisfy a uniform area bound. Our result is a natural generalization of the celebrated Schoen-Simon-Yau interior curvature estimates up to the free boundary. A direct corollary of our curvature estimates is a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For the case of $3$-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without any assumption on the area bound. This generalizes Schoen's interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

39 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the Riesz transform of the harmonic measure in a bounded uniform domain is bounded in O(L 2 ) by a scaling invariant condition on the Green function.
Abstract: Let $\Omega\subsetneq\mathbb R^{n+1}$ be open and let $\mu$ be some measure supported on $\partial\Omega$ such that $\mu(B(x,r))\leq C\,r^n$ for all $x\in\mathbb R^{n+1}$, $r>0$. We show that if the harmonic measure in $\Omega$ satisfies some scale invariant $A_\infty$ type conditions with respect to $\mu$, then the $n$-dimensional Riesz transform $$R_\mu f(x) = \int \frac{x-y}{|x-y|^{n+1}}\,f(y)\,d\mu(y)$$ is bounded in $L^2(\mu)$. We do not assume any doubling condition on $\mu$. We also consider the particular case when $\Omega$ is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.

36 citations


Journal ArticleDOI
TL;DR: In this paper, the Fourier-Mukai transform of a Brill-Noether locus of vector bundles is used to reconstruct the K3 surface of a curve of genus ε, such that ε is a composite number.
Abstract: Let $C$ be a curve of genus $g \geq 11$ such that $g-1$ is a composite number. Suppose $C$ is on a K3 surface whose Picard group is generated by the curve class $[C]$. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai's program to this situation: we show how to reconstruct the K3 surface containing the curve $C$ as a Fourier-Mukai transform of a Brill-Noether locus of vector bundles on $C$.

31 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that a certain graded ring derived from the logarithmic derivation module of a semisimple complex linear algebraic group is isomorphic to the Hessenberg variety.
Abstract: Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular semisimple Hessenberg variety $\mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $\mathcal{A}_I$ is isomorphic to $H^*(\mbox{Hess}(N,I))$ and $H^*(\mbox{Hess}(S,I))^W$, the invariants in $H^*(\mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of $W$ is isomorphic to the cohomology ring of the flag variety $G/B$. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map $H^*(G/B)\to H^*(\mbox{Hess}(N,I))$ announced by Dale Peterson and an affirmative answer to a conjecture of Sommers-Tymoczko are immediate consequences. We also give an explicit ring presentation of $H^*(\mbox{Hess}(N,I))$ in types $B$, $C$, and $G$. Such a presentation was already known in type $A$ or when $\mbox{Hess}(N,I)$ is the Peterson variety. Moreover, we find the volume polynomial of $\mbox{Hess}(N,I)$ and see that the hard Lefschetz property and the Hodge-Riemann relations hold for $\mbox{Hess}(N,I)$, despite the fact that it is a singular variety in general.

30 citations


Journal ArticleDOI
TL;DR: In this article, the authors define the expected degree G(k,n) of a real Grassmannian as the average number of real $k$-planes meeting nontrivially $k(n-k)$ random subspaces of the Grassmanian.
Abstract: We initiate the study of average intersection theory in real Grassmannians. We define the expected degree $\textrm{edeg} G(k,n)$ of the real Grassmannian $G(k,n)$ as the average number of real $k$-planes meeting nontrivially $k(n-k)$ random subspaces of $\mathbb{R}^n$, all of dimension $n-k$, where these subspaces are sampled uniformly and independently from $G(n-k,n)$. We express $\textrm{edeg} G(k,n)$ in terms of the volume of an invariant convex body in the tangent space to the Grassmanian, and prove that for fixed $k\ge 2$ and $n\to\infty$, $$ \textrm{edeg} G(k,n) = \textrm{deg} G_\mathbb{C}(k,n)^{\frac{1}{2} \epsilon_k + o(1)}, $$ where $\textrm{deg} G_\mathbb{C}(k,n)$ denotes the degree of the corresponding complex Grassmannian and $\epsilon_k$ is monotonically decreasing with $\lim_{k\to\infty} \epsilon_k = 1$. In the case of the Grassmannian of lines, we prove the finer asymptotic \begin{equation*} \textrm{edeg} G(2,n+1) = \frac{8}{3\pi^{5/2}\sqrt{n}}\, \left(\frac{\pi^2}{4} \right)^n \left(1+\mathcal{O}(n^{-1})\right). \end{equation*} The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set $X\subseteq\mathbb{R}\textrm{P}^{n-1}$ of dimension $n-k-1$ its Chow hypersurface $Z(X)\subseteq G(k,n)$, consisting of the $k$-planes $A$ in $\mathbb{R}^n$ whose projectivization intersects $X$. Denoting $N:=k(n-k)$, we show that $$ \mathbb{E}\#\left(g_1Z(X_1)\cap\cdots\cap g_N Z(X_N)\right) = \textrm{edeg} G(k,n) \cdot \prod_{i=1}^{N} \frac{|X_i|}{|\mathbb{R}\textrm{P}^{m}|}, $$ where each $X_i$ is of dimension $m=n-k-1$, the expectation is taken with respect to independent uniformly distributed $g_1,\ldots,g_m\in O(n)$ and $|X_i|$ denotes the $m$-dimensional volume of $X_i$.

29 citations


Journal ArticleDOI
TL;DR: In this paper, the intersection of two copies of Gr(2,5) embedded in ℙ9{{mathbb{P}}}^{9} was studied and it was shown that these intersections are deformation equivalent, derived equivalent Calabi-Yau threefolds.
Abstract: Abstract We study the intersection of two copies of Gr⁢(2,5){\\mathrm{Gr}(2,5)} embedded in ℙ9{{{\\mathbb{P}}}^{9}}, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi–Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi–Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi–Yau threefolds of the above type, which may be of independent interest.

28 citations


Journal ArticleDOI
TL;DR: In this article, a weaker notion related to the Kantorovich-Rubinstein transport distance, called K-random projection, has been proposed to provide linear extension operators for Lipschitz maps.
Abstract: Motivated by the notion of K-gentle partition of unity introduced in [12] and the notion of K-Lipschitz retract studied in [17], we study a weaker notion related to the Kantorovich-Rubinstein transport distance, that we call K-random projection. We show that K-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak* continuous operators. Finally we use this notion to characterize the metric spaces (X, d) such that the free space F(X) has the bounded approximation propriety.

27 citations


Journal ArticleDOI
TL;DR: In this article, the authors proved seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell and gave reductions of four other conjectures in terms of single-sum basic hypergeometric series.
Abstract: We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level $2$ modules of $A_9^{(2)}$. We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.

26 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proved uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations, which can be applied to study geometric regularity of singular solutions of complex MAME equations.
Abstract: We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein metrics on Kahler manifolds of nonnegative Kodaira dimensions.

Journal ArticleDOI
TL;DR: In this paper, the relative dynamical degrees of positive algebraic cycles are defined and a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semiconjugacy.
Abstract: Let $K$ be an algebraically closed field of arbitrary characteristic, $X$ an irreducible variety and $Y$ an irreducible projective variety over $K$, both are not necessarily smooth. Let $f:X\rightarrow X$ and $g:Y\rightarrow Y$ be dominant correspondences, and $\pi :X\rightarrow Y$ a dominant rational map such that $\pi \circ f=g\circ \pi$. We define relative dynamical degrees $\lambda _p(f|\pi )$ ($p=0,\ldots ,\dim (X)-\dim (Y)$). These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when $Y$ is smooth and $g$ is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy $(\varphi ,\psi )$ from $(X_2,f_2)\rightarrow (Y_2,g_2)$ to $(X_1,f_1)\rightarrow (Y_1,g_1)$ we have $\lambda _p(f_1|\pi _1)\geq \lambda _p(f_2|\pi _2)$ for all $p$. Many of our results are new even when $K=\mathbb{C}$. We make use of de Jong's alterations and Roberts' version of Chow's moving lemma. In the lack of resolution of singularities, the consideration of correspondences is necessary even when $f,g$ are rational maps. The case $K$ is not algebraically closed further requires working with correspondences over reducible varieties.

Journal ArticleDOI
TL;DR: In this article, it was shown that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close.
Abstract: Mean curvature flow of clusters of n-dimensional surfaces in R^{n+k} that meet in triples at equal angles along smooth edges and higher order junctions on lower dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.

Journal ArticleDOI
TL;DR: In this paper, the authors used non-commutative Iwasawa theory to investigate the values at zero of higher derivatives of the p-adic Artin L-series.
Abstract: Abstract We use techniques of non-commutative Iwasawa theory to investigate the values at zero of higher derivatives of p-adic Artin L-series.

Journal ArticleDOI
TL;DR: In this article, it was shown that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in ℙg+r{mathbb{P}€ g+r}, not a cone.
Abstract: Abstract Let C be a smooth projective curve (resp. (S,L){(S,L)} a polarized K⁢3{K3} surface) of genus g⩾11{g\\geqslant 11}, with Clifford index at least 3, considered in its canonical embedding in ℙg-1{\\mathbb{P}^{g-1}} (resp. in its embedding in |L|∨≅ℙg{|L|^{\\vee}\\cong\\mathbb{P}^{g}}). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in ℙg+r{\\mathbb{P}^{g+r}}, not a cone, with dim⁡(Y)=r+2{\\dim(Y)=r+2} and ωY=𝒪Y⁢(-r){\\omega_{Y}=\\mathcal{O}_{Y}(-r)}, if the cokernel of the Gauss–Wahl map of C (resp. H1⁡(TS⊗L∨){\\operatorname{H}^{1}(T_{S}\\otimes L^{\\vee})}) has dimension larger than or equal to r+1{r+1} (resp. r). This relies on previous work of Wahl and Arbarello–Bruno–Sernesi. We provide various applications.

Journal ArticleDOI
TL;DR: In this paper, an infinite-dimensional Lyapunov-Schmidt reduction method was used to reduce the fractional Yamabe problem to an infinite dimensional Toda type system.
Abstract: We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.


Journal ArticleDOI
TL;DR: In this paper, the minimal model theory for projective klt pairs of dimension at most n was established under the assumption that the log canonical divisor is relatively log abundant.
Abstract: Under the assumption of the minimal model theory for projective klt pairs of dimension $n$, we establish the minimal model theory for lc pairs $(X/Z,\Delta)$ such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most $n$. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.

Journal ArticleDOI
TL;DR: In this article, a new proof of Kirchberg's stable classification theorem was presented, which states that two separable, nuclear, stable/unital, α-algebras are isomorphic if and only if their ideal lattices are order isomorphic, or equivalently, their primitive ideal spaces are homeomorphic.
Abstract: I present a new proof of Kirchberg’s O 2 -stable classification theorem: two separable, nuclear, stable/unital, O 2 -stable C ∗ -algebras are isomorphic if and only if their ideal lattices are order isomorphic, or equivalently, their primitive ideal spaces are homeomorphic. Many intermediate results do not depend on pure infiniteness of any sort.

Journal ArticleDOI
TL;DR: In this article, it was shown that half spaces are the only stable nonlocal s-minimal cones in ℝ 3 √ R 3 for s ∈ ( 0, 1 ) {s\in(0, 1)} sufficiently close to 1.
Abstract: Abstract We prove that half spaces are the only stable nonlocal s-minimal cones in ℝ 3 {\mathbb{R}^{3}} , for s ∈ ( 0 , 1 ) {s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from s = 1 {s=1} . In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

Journal ArticleDOI
TL;DR: In this article, the trace properties of non-local minimal graphs with a graphical structure were studied, and it was shown that at any boundary point at which the trace from inside happens to coincide with the exterior data, also the tangent planes of the traces necessarily coincide with those of the exterior datum.
Abstract: The main goal of this article is to understand the trace properties of nonlocal minimal graphs in~$\R^3$, i.e. nonlocal minimal surfaces with a graphical structure. We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum. This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error. We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.

Journal ArticleDOI
TL;DR: In this paper, the authors introduced a new definition of the Dehn function and used it to prove several theorems, including the existence and regularity of area-minimizing Sobolev mappings in metric spaces.
Abstract: Abstract The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.

Journal ArticleDOI
TL;DR: In this article, the counting function of common zeros of two meromorphic functions in various contexts is studied and a general version of a conjectural "asymptotic gcd" inequality of Pasten and the second author is proved.
Abstract: We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural "asymptotic gcd" inequality of Pasten and the second author, and consider moving targets versions of our results.

Journal ArticleDOI
TL;DR: A trace on a C*-algebra is amenable if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm) as discussed by the authors.
Abstract: A trace on a C*-algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm). Using that the double commutant of a nuclear C*-algebras is hyperfinite, it is easy to see that traces on nuclear C*-algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear C*-algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of C*-algebras and, in particular, using a version of the Weyl-von Neumann Theorem due to Elliott and Kucerovsky.

Journal ArticleDOI
TL;DR: For a classical group over a non-archimedean local field of odd residual characteristic p, this paper constructed all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cusidal type.
Abstract: For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.

Journal ArticleDOI
TL;DR: In this article, the authors use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals.
Abstract: Abstract In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to commutative algebra.

Journal ArticleDOI
TL;DR: In this paper, the boundary capacity of mean-convex fill-ins with nonnegative scalar curvature and singular metrics with singular metrics has been studied for Riemannian Schwarzschild manifolds.
Abstract: We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown--York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.


Journal ArticleDOI
TL;DR: In this article, the authors give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map.
Abstract: We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.

Journal ArticleDOI
TL;DR: In this article, it was shown that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its topology and compute it.
Abstract: We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu. (We do not use a Lusternik-Schnirelmann type theorem on the existence of at least three simple closed geodesics.)