The first eigenvalue of embedded minimal hypersurfaces in the unit sphere

Abstract: In this article, we prove that for an embedded minimal hypersurface $Σ<sup>{m}$ in $S<sup>{m+1}$, the first eigenvalue $λ<em>1$ of the Laplacian operator on $Σ$ satisfies: $$λ_1&gt; \frac{m}{2}+G(m, |A|</em>{\max}, |A|<em>{\min} ) ,$$ where $|A|</em>{\max}$ and $|A|<em>{\min}$ denote the maximum and minimum of the norm of the second fundamental form on $Σ$, respectively; $G(m, |A|</em>{\max}, |A|<em>{\min} )$ is a positive constant that depends only on $m,|A|</em>{\max}, |A|_{\min}$. In particular, when the norm $|A|$ of the second fundamental form is constant, we can obtain a gap depending only on $m$, i.e., $$λ_1&gt;\frac{m}{2} \left(1+ c \right) ,$$ where $c$ is a positive absolute constant. This improves the previous result of Choi and Wang \cite{chw1983first}, which gave $λ_1\geq \frac{m}{2}$. Our result shows that one can skip proving Chern's conjecture to directly improve Choi-Wang's result. This also generalizes Tang and Yan's work \cite{tangyan2013isoparametric}. Based on the proof of the result above, using the lower bound of the Steklov eigenvalue, we prove that if the norm $|A|$ of the second fundamental form is constant, then $$|A| \leq \frac{C(m)\textup{Volume}(Σ)}{\textup{Volume}(S<sup>m)},$$ where $C(m)$ is a constant that depends only on $m$. This provides a uniform estimate for the scalar curvature of embedded minimal hypersurfaces with constant norm of the second fundamental form. Moreover, this may useful for Chern's problem.

• The paper improves the Choi–Wang estimate from λ₁ ≥ m/2 to an explicit strict lower bound whose gap remains nonzero as maximum curvature grows.
• The proof uses higher-order Reilly identities, third-derivative estimates, tubular-neighborhood traces, and boundary curvature inequalities to retain information discarded in earlier arguments.
• For constant second-fundamental-form norm, the result gives λ₁ > (1/2+c)m with an asymptotic gap near 0.042m and connects Steklov bounds to volume-based curvature control.

Overview and main results

This paper by Yuhang Zhao establishes a quantitative improvement of the Choi–Wang lower bound for the first Laplacian eigenvalue of embedded minimal hypersurfaces in the unit sphere. For a compact minimal embedding $F:\Sigma^m \to S^{m+1}$ that is not totally geodesic, the main theorem asserts

$$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$

where $|A|_{\max}$ and $|A|_{\min}$ are the maximum and minimum of the norm of the second fundamental form. The bracketed quantity is positive, so this strictly improves the classical bound $\lambda_1 \geq m/2$ of Choi and Wang (2603.20890), which had remained the best general result toward Yau's conjecture ($\lambda_1 = m$) despite a sequence of refinements by Zhao (Zhao, 2023), Duncan–Spruck–Sire, and Jiménez–Tapia–Zhou (Jiménez et al., 2024). A key structural point is that all previous gaps decay to zero as $|A|_{\max} \to \infty$, and doublings provide infinitely many examples with arbitrarily large $|A|_{\max}$; the new estimate retains a nonzero gap uniformly in $|A|_{\max}$.

In the special case where $|A|$ is constant (so $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$0), the gap becomes dimension-dependent only:

$$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$1

with an absolute constant $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$2; asymptotically as $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$3 the gap is approximately $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$4. This is significant because Tang and Yan proved Yau's conjecture for isoparametric hypersurfaces only via their classification; here the author shows one can bypass Chern's conjecture entirely and still improve Choi–Wang under constant $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$5, using only basic properties of minimal hypersurfaces.

Method: higher-order Reilly computation

The proof departs from Choi–Wang at a specific point. Their argument applies Reilly's formula to harmonic extensions $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$6 of a first eigenfunction $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$7 on the two regions $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$8 cut out by $$\lambda_1 > \frac{m}{2} + \frac{\sqrt{m^2-1}}{48|A|_{\max}}\left[\left(\frac{10}{\sqrt{11}}-\sqrt{\frac{m+1}{m}}\right)|A|_{\min} + \frac{13}{2\sqrt{11}}\sqrt{m} - \frac{m+2}{\sqrt{m+1}}\right],$$9, obtaining

$|A|_{\max}$0

and discards the Hessian term. The present work instead computes $|A|_{\max}$1 and $|A|_{\max}$2 explicitly. Exploiting the ambient sphere, the author applies Reilly's formula to each component function $|A|_{\max}$3 rather than performing a general tensorial Bochner computation, then integrates over both regions and adds the results so that boundary terms cancel. The resulting identity (Theorem 3 of the paper) relates third-order interior integrals $|A|_{\max}$4 to boundary integrals involving $|A|_{\max}$5, the normal derivatives of $|A|_{\max}$6 and $|A|_{\max}$7, and the pullback Hessians $|A|_{\max}$8, $|A|_{\max}$9.

Two auxiliary inputs complete the scheme:

• Tubular neighborhood trace estimate: using Howard's rolling theorem, the rolling radius equals the focal distance $|A|_{\min}$0, bounded below via Simons' inequality $|A|_{\min}$1. A cutoff-function argument on the tubular neighborhood yields $|A|_{\min}$2 with $|A|_{\min}$3 arbitrarily small.
• Boundary algebra: pointwise identities on $|A|_{\min}$4 decompose $|A|_{\min}$5 into pieces involving $|A|_{\min}$6, the sums $|A|_{\min}$7, differences of tangential Hessian data, and the term $|A|_{\min}$8. Young-type inequalities with parameters $|A|_{\min}$9 and a free positive function $\lambda_1 \geq m/2$0 absorb the cross terms; the constraint $\lambda_1 \geq m/2$1 (a Lagrange multiplier extremum under $\lambda_1 \geq m/2$2, $\lambda_1 \geq m/2$3) controls the quartic curvature terms.

Choosing $\lambda_1 \geq m/2$4 and minimizing a one-variable concave function $\lambda_1 \geq m/2$5 over $\lambda_1 \geq m/2$6 produces the final explicit gap. The author notes candidly that several estimates in the chain are rough, and poses whether the gap can be pushed close to $\lambda_1 \geq m/2$7.

A Steklov eigenvalue bound and uniform scalar curvature estimate

The second half of the paper derives a volume-controlled upper bound on $\lambda_1 \geq m/2$8 when it is constant. From the main computation with a different choice of $\lambda_1 \geq m/2$9 and $\lambda_1 = m$0, one obtains

$\lambda_1 = m$1

The left side dominates the first nonzero Steklov eigenvalue $\lambda_1 = m$2 of the summed Dirichlet-to-Neumann map $\lambda_1 = m$3. The paper proves

$\lambda_1 = m$4

where $\lambda_1 = m$5. The proof glues the two harmonic extensions into a globally Lipschitz function on $\lambda_1 = m$6, establishes a mean value formula valid for Lipschitz functions — requiring an additional boundary integral over $\lambda_1 = m$7 absent from the smooth case — and closes via the monotonicity estimate $\lambda_1 = m$8, adapted from Brendle and Colding–Minicozzi to the spherical setting. Only volume growth of $\lambda_1 = m$9 enters, not finer geometry.

Combining the two bounds gives, for constant $|A|_{\max} \to \infty$0,

$|A|_{\max} \to \infty$1

with $|A|_{\max} \to \infty$2 expressed explicitly through $|A|_{\max} \to \infty$3 and $|A|_{\max} \to \infty$4. Together with the area bound $|A|_{\max} \to \infty$5, this also yields two-sided control relating $|A|_{\max} \to \infty$6 and volume. The implication for Chern's problem — whether $|A|_{\max} \to \infty$7 universally when $|A|_{\max} \to \infty$8 is constant — is direct: under the embedding hypothesis, a uniform volume upper bound would resolve it. The author frames volume as the more tractable quantity, though no such volume bound is established here.

Limitations and open questions

Several restrictions qualify the results. The eigenvalue theorem requires embeddedness throughout; immersed examples fall outside its scope. The gap, while uniform in $|A|_{\max} \to \infty$9, shrinks like $|A|_{\max}$0 unless $|A|_{\max}$1 is comparable to $|A|_{\max}$2, and the constant-$|A|_{\max}$3 case yields only the modest additive term $|A|_{\max}$4 rather than anything approaching $|A|_{\max}$5 — the author attributes this to rough intermediate estimates and to the limited structural knowledge of constant-$|A|_{\max}$6 hypersurfaces in dimensions above four. The curvature bound depends on the unresolved question of uniform volume control, and the Steklov comparison between $|A|_{\max}$7 and $|A|_{\max}$8 is left open except when dependence on $|A|_{\max}$9 is permitted, where Colbois–Girouard–Hassannezhad already suffices.

Conclusion

The paper converts the discarded Hessian term in the Choi–Wang argument into a genuine spectral gap, yielding the first improvement of $|A|_{\max}$0 that survives as $|A|_{\max}$1 and a dimension-only gap in the constant-$|A|_{\max}$2 case without recourse to isoparametric classification. Its secondary contribution — a Steklov lower bound leading to $|A|_{\max}$3 — reduces Chern's problem, under embeddability, to a uniform volume estimate. Whether the eigenvalue gap can be sharpened toward $|A|_{\max}$4, and whether the required volume bound holds, remain open.