Simplifies Riemann geometry computations without indices.
problem Complex computations in Riemann geometry.
method Uses forms with Clifford algebra values and a specific group action.
result Simplified computations in Riemann geometry.
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Estimates the growth of Morse index for free boundary minimal hypersurfaces.
problem Estimating the Morse index of free boundary minimal hypersurfaces.
method Adapting Song's method to the free boundary case.
result The Morse index grows linearly with the sum of Betti numbers.
Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing th…
Develops obstruction theory for a specific 4-manifold index.
problem Computing the -index of 4-manifolds with free involution.
method Uses spectral sequences and cohomology with twisted coefficients.
result Computes the -index for various examples.
Study shows infinite kernels in topological monodromy for curve families.
problem Understanding kernels of topological monodromy representations.
method Extending Kuno's arguments and using Carlson-Toledo techniques.
result Kernels are infinite for certain linear systems on surfaces.
The paper proves bounds on the Morse index of free boundary minimal hypersurfaces.
problem Finding bounds on the Morse index of free boundary minimal hypersurfaces.
method Min-max theory applied to -dimensional compact manifolds with boundary.
result Establishes general upper bounds for the Morse index of free boundary minimal hypersurfaces.
Paper proves rigidity and index of Y-cones in unit ball.
problem Proving rigidity and index of Y-cones in the unit ball.
method Analyzes conformal and minimal immersions of Y-cones meeting the boundary orthogonally.
result Y-cones are the only free boundary minimal surfaces with Morse index 2(n-2).
The paper extends Hodge-de Rham and Lichnérowicz Laplacians to double forms and proves vanishing theorems.
problem Extending Laplacians to double forms and proving vanishing theorems.
method Introduced a new product on double forms to establish index-free formulas for curvature terms in Weitzenböck formulas for , , and . Proved vanishing theorems for and on symmetric double forms.
result Vanishing theorems for the Hodge-de Rham Laplacian and on symmetric double forms.