Junya Takahashi's Homepage

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Lectures

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Research interests

Spectral Geometry is to reveal the geometric information of Riemannian manifolds from the spectrum of elliptic differential operators obtained from Riemannian manifolds. In particular, when dealing with the Hodge-Laplacian acting on differential forms as an elliptic differential operator, we can also get the topological information of the considering manifold from its spectrum.
Now, when a Riemannian manifold collapses or degenerates, how is its spectrum influenced ?
I am studying the behavior and the limit of the spectrum under such deformations, and also the relationship between manifolds and differential operators.
This research field is very interesting and has a lot of interaction with differential geometry, topology, singularity theory, and analysis of partial differential equations, differential operators and elliptic boundary value problems.
  1. Spectral Geometry (the eigenvalues of the Hodge-Laplacian acting on p-forms)
  2. Spectrum and Collapsing of Riemannian manifolds (small and large eigenvalues)
  3. Geometry, topology and analysis of differential forms (L^2-Stokes theorem, L^2-harmonic forms and L^2-cohomology)
  4. Analysis on manifolds and singular spaces (elliptic boundary value problems, resolution of singularities)

Intérêts de recherches

  1. géométrie spectrale (les valeurs propres du laplacien agissant sur les p-formes différentielles)
  2. spectre et effondrements de variétés riemanniennes (des petites et des grands valeurs propres quand les variétés s'effondre)
  3. géométrie, topologie et analyse de formes différentielles (théorème de L^2-Stokes et des formes harmoniques L^2)
  4. analyse sur des variétés et sur des espaces singularitiés (problèmes aux limites elliptiques, resolution de singularités)

Published Papers

  1. (with Colette Anné), Small eigenvalues of the rough and Hodge Laplacians under fixed volume, pdf,
    Ann. Fac. Sci Toulouse, 33 (2024), 123-151.
  2. L^2-harmonic forms on incomplete Riemannian manifolds with positive Ricci curvature, pdf,
    Mathematics 6 (5), 75 (2018); doi: 10.3390/math6050075.
  3. (with Colette Anné), Partial collapsing and the spectrum of the Hodge-de Rham operator, pdf,
    Analysis & PDE. 8 (2015), 1025-1050.
  4. (with Colette Anné), p-spectrum and collapsing of connected sums, pdf,
    Trans. Amer. Math. Soc. 364 (2012), 1711-1735.
  5. Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian, pdf,
    Manuscripta Math. 121 (2006), 191-200.
  6. The gap of the eigenvalues for p-forms and harmonic p-forms of constant length, pdf,
    J. Geom. Phys. 54 (2005), 476-484.
  7. Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms, pdf,
    Math. Zeit. 250 (2005), 43-57.
  8. On the gap between the first eigenvalues of the Laplacian on functions and p-forms, pdf,
    Ann. Global Anal. Geom. 23 (2003), 13-27.
  9. Small eigenvalues on p-forms for collapsings of the even-dimensional spheres, pdf,
    Manuscripta Math. 109 (2002), 63-71.
  10. Collapsing of connected sums and the eigenvalues of the Laplacian, pdf,
    J. Geom. Phys. 40 (2002), 201-208.
  11. On the gap between the first eigenvalues of the Laplacian on functions and 1-forms, pdf,
    J. Math. Soc. Japan 53 (2001), 307-320.
  12. Upper bounds for the eigenvalues of the Laplacian on forms on certain Riemannian manifolds, pdf,
    J. Math. Sci. Univ. Tokyo 6 (1999), 87-99.
  13. The first eigenvalue of the Laplacian on p-forms and metric deformations, pdf,
    J. Math. Sci. Univ. Tokyo 5 (1998), 333-344.

Preprints

  1. (with Colette Anné), Small eigenvalues of the Hodge-Laplacian with sectional curvature bouded below, preprint, (2024).

Research Visiting

Links

last update: 13 December 2024

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