GLOBAL ANALYSIS · GEOMETRY · PDE

Research & publications

My work focuses on developing new intrinsic results in global analysis, with emphasis on problems that connect different areas of mathematics: differential geometry, functional analysis, Sobolev theory, partial differential equations, vector bundles, and variational methods.

I am especially interested in geometric formulations that make it possible to study PDE problems on Riemannian manifolds and vector bundles while keeping both the local structure and the global phenomena visible.

View publications Research directions

A section in ℝ³ — One point in each fiber; together, a section.

Main directions

Sobolev spaces on manifolds and vector bundles

I study intrinsic definitions of Sobolev spaces, norm equivalence, density results, Sobolev embeddings, Rellich–Kondrashov compactness, and trace theory in geometric settings.

A trace at the boundary — The highlighted circles approach the boundary.

Geometric integration by parts

A central direction of my work is the formulation of higher-order integration by parts identities on Riemannian manifolds with boundary and on vector bundles, including boundary terms and formal adjoints of covariant derivatives.

A search for critical pointsγ̇ = −∇J

Two trajectories descend toward the minima.

PDE on vector bundles

I am currently working on nonlinear elliptic problems on vector bundles, especially quasilinear equations and ground state solutions through variational methods.

Bounded geometry and noncompact analysis

I am interested in uniform estimates on noncompact manifolds, including bounded geometry, trace theorems, local-to-global constructions, and fractional Sobolev spaces.

Topology and realcompact spaces

I also maintain an interest in general topology, especially realcompact spaces and related extensions.

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}
}