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ASYMPTOTIC CURVATURE OF MODULI SPACES FOR CALABI–YAU THREEFOLDS
, 902
"... Abstract. Motivated by the classical statements of Mirror Symmetry, we study certain Kähler metrics on the complexified Kähler cone of a Calabi– Yau threefold, conjecturally corresponding to approximations to the Weil– Petersson metric near large complex structure limit for the mirror. In particular ..."
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Abstract. Motivated by the classical statements of Mirror Symmetry, we study certain Kähler metrics on the complexified Kähler cone of a Calabi– Yau threefold, conjecturally corresponding to approximations to the Weil– Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemannian metric (defined via cupproduct) on a level set of the Kähler cone is seen to be analogous to a slice of the Weil– Petersson metric near large complex structure limit. This enables us to give counterexamples to a conjecture of Ooguri and Vafa that the Weil–Petersson metric has nonpositive scalar curvature in some neighbourhood of the large complex structure limit point.
On a class of threedimensional integrable Lagrangians
"... We characterize nondegenerate Lagrangians of the form f(ux, uy, ut) dx dy dt such that the corresponding EulerLagrange equations (fux)x +(fuy)y +(fut)t = 0 are integrable by the method of hydrodynamic reductions. The integrability conditions constitute an overdetermined system of fourth order PDE ..."
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Cited by 4 (1 self)
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We characterize nondegenerate Lagrangians of the form f(ux, uy, ut) dx dy dt such that the corresponding EulerLagrange equations (fux)x +(fuy)y +(fut)t = 0 are integrable by the method of hydrodynamic reductions. The integrability conditions constitute an overdetermined system of fourth order PDEs for the Lagrangian density f, which is in involution and possess interesting differentialgeometric properties. The moduli space of integrable Lagrangians, factorized by the action of a natural equivalence group, is threedimensional. Familiar examples include the dispersionless KadomtsevPetviashvili (dKP) and the BoyerFinley Lagrangians, f = u 3 x/3 + u 2 y − uxut and f = u 2 x + u 2 y − 2e ut, respectively. A complete description of integrable cubic and quartic Lagrangians is obtained. Up to the equivalence transformations, the list of integrable cubic Lagrangians reduces to three examples, f = uxuyut, f = u 2 xuy + uyut and f = u 3 x/3 + u 2 y − uxut (dKP). There exists a unique integrable quartic Lagrangian, f = u 4 x + 2u 2 xut − uxuy − u 2 t. We conjecture that these examples exhaust the list of integrable polynomial Lagrangians which are essentially threedimensional (it was verified that there exist no polynomial integrable Lagrangians of degree five). We prove that the EulerLagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless ‘Lax pair’.
SOME GEOMETRY AND COMBINATORICS FOR THE SINVARIANT OF TERNARY CUBICS.
, 2005
"... In earlier papers [5,4], the Sinvariant of a ternary cubic f was related to the curvature of the level set f = 1 in R3. In particular, when f arises from the cubic form on the second cohomology of a smooth projective threefold with second Betti number three, the value of the Sinvariant is closely ..."
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Cited by 2 (2 self)
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In earlier papers [5,4], the Sinvariant of a ternary cubic f was related to the curvature of the level set f = 1 in R3. In particular, when f arises from the cubic form on the second cohomology of a smooth projective threefold with second Betti number three, the value of the Sinvariant is closely linked to the behaviour of this curvature on the open subset of this level set consisting of Kähler classes [5]. In this paper, we consider the cubic forms arising from complete intersections in the product of three projective spaces, and investigate various conjectures of a combinatorial nature suggested concerning their invariants.
Higherdimensional black hole Geometric Thermodynamics
"... Abstract. The pseudoRiemannian Geometry is utilized in Mathematical Optimization, Thermodynamics or in Statistics as an important tool for recent research. Given a pseudoRiemannian manifold (M, g) and a smooth function f: M → R, whose Hessian with respect to g is nondegenerate, one can define on ..."
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Abstract. The pseudoRiemannian Geometry is utilized in Mathematical Optimization, Thermodynamics or in Statistics as an important tool for recent research. Given a pseudoRiemannian manifold (M, g) and a smooth function f: M → R, whose Hessian with respect to g is nondegenerate, one can define on M the associated pseudoRiemannian Hessian metric h = Hessgf. In the following, we apply this methodology for describing the geometrical properties of some interesting mathematical objects like the higher dimensional ReissnerNördstrom black holes. The paper is organized as follows. Section 1 reviews the history of pseudoRiemannian Geometry and points out which Hessian is more convenient for physical problems. Section 2 gives the Christoffel symbols and the system of geodesics of pseudoRiemannian manifold (M,h = Hessgf), establishes the relation between the components of the curvature tensors field of (M,h) and (M, g), and determines the PDEs representing the coincidence between the Christoffel symbols of (M,h) and the Christoffel symbols of (M, g). The last section presents the comparison of nulllength curves trajectories obtained with the two metrics for a 5dimensional RN black hole.
ON WARPED PRODUCT SPACETIMES: CONFORMAL HYPERBOLICITY AND KILLING VECTOR FIELDS
, 2006
"... Abstract. In the first part of this paper, we investigate the conformal hyperbolicity and conjugate points of standard static spacetimes. Moreover, an upper bound for the timelike diameter of a standard static spacetime is also obtained by the Ricci tensor inequalities. In the second part, we cons ..."
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Abstract. In the first part of this paper, we investigate the conformal hyperbolicity and conjugate points of standard static spacetimes. Moreover, an upper bound for the timelike diameter of a standard static spacetime is also obtained by the Ricci tensor inequalities. In the second part, we consider Killing vector fields on standard static spacetimes and obtain equations for a vector field on a standard static spacetime to be Killing. We also provide a characterization of Killing vector fields on standard static spacetimes with compact fibers.
Some geometry and combinatorics for the Sinvariant of ternary cubics. P.M.H. Wilson
, 2004
"... Introduction. Given a real ternary cubic form f(x1, x2, x3), there is a pseudoRiemannian metric, given by the matrix (gij) = 1 6 (∂2 f/∂xi∂xj), defined on the open subset of R 3 where the determinant h = det(gij) is nonzero. Building on previous work in [5,4], we determine ..."
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Introduction. Given a real ternary cubic form f(x1, x2, x3), there is a pseudoRiemannian metric, given by the matrix (gij) = 1 6 (∂2 f/∂xi∂xj), defined on the open subset of R 3 where the determinant h = det(gij) is nonzero. Building on previous work in [5,4], we determine