# P4024: Heat kernel geometry and Gromov's volume growth conjecture

- ID: `P4024`
- Reference: `heat-kernel-geometry-and-gromov-s-volume-growth-conjecture`
- Page: https://theoremdb.org/statements/P4024
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/heat-kernel-geometry-and-gromov-s-volume-growth-conjecture?representation=markdown
- Record maturity: Reviewed problem

## The problem

Determine whether every complete noncompact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and scalar curvature at least one satisfies: \[ \Vol_g (B(p, R))\le C_{n}R^{n-2} \] for all $p\in M$ and $R>0$.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `heat-kernel-geometry-and-gromov-s-volume-growth-conjecture`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.