Publications

Articles and preprints

Geometric Integration by Parts and Sobolev Spaces on Vector Bundles: A Unified Global Approach

Carlos Daniel Velázquez-Mendoza and María de los Ángeles Sandoval-Romero.
arXiv:2602.01016, 2026.

This work develops an intrinsic framework for Sobolev spaces on vector bundles over Riemannian manifolds. Its central result is a higher-order geometric integration by parts formula characterizing the formal adjoint of the covariant derivative as a global differential operator. The article also establishes a unified approach to weak derivatives, Sobolev structures, and geometric differential operators in the context of vector bundles.

arXiv PDF DOI

Global Analysis Sobolev Spaces Vector Bundles Geometric PDE

A Simple Proof of Metric Independence of Sobolev Norms on Compact Manifolds

Carlos D. Velázquez-Mendoza and María de los Ángeles Sandoval-Romero.
In Analysis and PDE in Latin America, Trends in Mathematics, vol. 15, Birkhäuser, Cham, 2026, pp. 125–133.

This chapter presents a detailed proof of the metric independence of Sobolev norms on compact Riemannian manifolds. The exposition emphasizes the intrinsic geometric structure underlying Sobolev spaces and provides a concise argument accessible to graduate students and researchers working in geometric analysis.

Springer DOI

Sobolev Norms Compact Manifolds Riemannian Metrics

Work in progress

Ground state solutions for quasilinear equations on vector bundles

Carlos Daniel Velázquez Mendoza, María de los Ángeles Sandoval Romero, and Romulo Diaz Carlos.
Work in progress.

This project studies ground state solutions for quasilinear elliptic equations on vector bundles over compact Riemannian manifolds with boundary. If \(u\) is a section of a vector bundle \(E\to M\) endowed with a fiber metric and a compatible connection, the natural first-order object is the covariant derivative \(\nabla^{E}u\). The corresponding divergence-form operator must be formulated through the formal adjoint of \(\nabla^{E}\), rather than through the scalar Riemannian divergence alone. This leads to a geometric variational framework for equations modeled by expressions of the form \[ (\nabla^{E})^{*}\left(a(|\nabla^{E}u|^{p})|\nabla^{E}u|^{p-2}\nabla^{E}u\right)=f(u), \] together with Dirichlet-type boundary conditions.

Variational Methods Quasilinear Equations Vector Bundles Ground States

BibTeX

@misc{velazquez_sandoval_2026_geometric_integration,
  title = {Geometric Integration by Parts and Sobolev Spaces on Vector Bundles: A Unified Global Approach},
  author = {Velázquez-Mendoza, Carlos Daniel and Sandoval-Romero, María de los Ángeles},
  year = {2026},
  eprint = {2602.01016},
  archivePrefix = {arXiv},
  primaryClass = {math.AP},
  doi = {10.48550/arXiv.2602.01016}
}

@incollection{velazquez_sandoval_2026_metric_independence,
  title = {A Simple Proof of Metric Independence of Sobolev Norms on Compact Manifolds},
  author = {Velázquez-Mendoza, Carlos D. and Sandoval-Romero, María de los Ángeles},
  booktitle = {Analysis and PDE in Latin America},
  series = {Trends in Mathematics},
  volume = {15},
  publisher = {Birkhäuser},
  address = {Cham},
  year = {2026},
  pages = {125--133},
  doi = {10.1007/978-3-031-99557-6_16}
}