# An integral of the Hodge-Neumann Laplacian on a Riemannian manifold

## Background

Let $$M$$ be a compact, oriented Riemannian manifold with boundary, with $$\text{vol}$$ the Riemannian volume form. Let $$\nu$$ be the outwards unit normal vector field on $$\partial M$$ and $$\nu^\flat$$ its dual 1-form (under musical isomorphism). Then let $$\text{vol}_\partial=\iota_\nu \text{vol}$$ orient $$\partial M$$ (standard). The Riemannian metric induces an $$L^2$$ structure on $$\Omega^k$$ (this is the dot product for differential forms, which differs from the dot product for tensor fields by a factor of $$\frac{1}{k!}$$).

Let $$\jmath:\partial M \to M$$ be the inclusion map. Then we can define the tangential and normal part of a differential form near the boundary, following Günter Schwarz $$\forall w \in \Omega^k(M): \mathbf{t}w = \iota_\nu(\nu^\flat\wedge w), \mathbf{n}w = w - \mathbf{t}w = \nu^\flat\wedge\iota_\nu w$$ Obviously $$\mathbf{n}f=0 \; \forall f\in \Omega^0(M)=C^\infty(M)$$ and $$\star$$ (Hodge star) is $$L^2$$ isomorphism with $$\star \mathbf{n} = \mathbf{t} \star,\; \mathbf{n} \star= \star \mathbf{t}$$.

Now define $$\Omega^k_{\text{hom}N}(M)=\{w\in \Omega^k: \mathbf{n}w=0,\mathbf{n}dw=0\}$$ (relative Neumann boundary condition), and $$\mathcal{H}^k_N =\{w\in \Omega^k: \mathbf{n}w=0,dw=0,\delta w=0\}$$ (Neumann harmonic fields), where $$\delta$$ is the codifferential. Let $${(\mathcal{H}^k_N)}^{\perp}$$ be the $$L^2$$ orthogonal complement in $$\Omega^k$$.

Let $$\Delta=-(d\delta +\delta d)$$ be the usual Hodge Laplacian (with negative spectrum). Then if we restrict its domain, we get the Hodge-Neumann Laplacian $$\Delta_N : \Omega^k_{\text{hom}N}(M) \cap {(\mathcal{H}^k_N)}^{\perp}\to {(\mathcal{H}^k_N)}^{\perp}$$ which is $$C^\infty$$ homeomorphism and essentially $$L^2$$-self-adjoint. The inverse is called the Neumann potential, which is a compact operator on $${(\mathcal{H}^k_N)}^{\perp}$$ (subspace of $$L^2$$), so there is an orthonormal basis consisting of eigenvectors (spectral theorem for compact operators). See Günter Schwarz for more details.

The formulas for integration by parts are:

$$\ll du,v \gg_M = \ll u,\delta v \gg_M + \ll \jmath^* u, \jmath^* \iota_\nu v \gg_{\partial M}$$ $$\mathcal{D}(u,v)= \ll -\Delta u,v \gg_M + \ll \jmath^* \iota_\nu du, \jmath^* v \gg_{\partial M} - \ll \jmath^* \delta u, \jmath^* \iota_\nu v \gg_{\partial M}$$ where $$\ll \cdot, \cdot \gg_M$$ is the $$L^2$$ product on $$M$$ and $$\mathcal{D}(u,v) = \ll du, dv \gg_M + \ll \delta u, \delta v \gg_M$$ is called the Dirichlet integral. On $$D(\Delta_N)$$, we also have the Poincare-Hodge inequality (consider $$0 \in \mathbb{N}$$)

$$||w||_{W^{s+1,p}} \sim ||dw||_{W^{s,p}} + ||\delta w||_{W^{s,p}} \; \forall s\in \mathbb{N}, \forall p \in [2,+\infty)$$

## Question

Let $$w \in \Omega^k$$ and $$w \in D(\Delta_N^K)\; \forall K \in \mathbb{N}$$. Can we show there exists $$C_{k,K,M}\geq 0$$ such that $$\int_M |w|^{2K} \Delta(|w|^2) \leq C_{k,K,M} \int_M |w|^{2K+2} \; \forall K \in \mathbb{N}$$

where $$|.|$$ is the fiber norm induced by the Riemannian metric (again, the differential forms version differs from the tensor version by a factor of $$\frac{1}{k!}$$). If this is true, it would have non-trivial implications for index theory, Hodge theory and fluid dynamics (being a form of dissipativity). It is certainly true (in fact $$C_{k,K,M}=0$$) when $$w$$ is a scalar function ($$w=f$$,$$\partial_\nu f =0$$) or $$\partial M = \emptyset$$, or that the metric is flat. So already most possible cases for PDE are covered. I want to know if this is true for general M. If that fails, how about $$\ll |w|^{2K}w, \Delta w \gg_M \leq C_{k,K,M} \int_M |w|^{2K+2} \; \forall K \in \mathbb{N}$$ (which is a weaker statement, following from the previous by Bochner's formula)

## My attempt

$$\mathbf{n}w=0$$ and $$\mathbf{n}dw=0$$ are equivalent to $$\iota_\nu w = 0$$ and $$\iota_\nu dw = 0$$. Integration by parts immediately yields

\begin{align} \ll \Delta(|w|^2), |w|^{2K} \gg_M & = - \ll d(|w|^2), d(|w|^{2K}) \gg_M + \ll \partial_\nu(|w|^2), |w|^{2K} \gg_{\partial M} \\ & \leq \; \ll \partial_\nu(|w|^2), |w|^{2K} \gg_{\partial M} = 2 \ll \nabla_\nu w, |w|^{2K} w \gg_{\partial M} \\ & = \frac{2}{K+1} \int_{\partial M} \partial_\nu(|w|^{2(K+1)}) = \frac{2}{K+1} \int_M \Delta(|w|^{2(K+1)}) \end{align}

The last 2 equalities look nice (for future applications), but I'm not sure if they are useful here.

We shall use Penrose's abstract index notation (also confer Wald's General Relativity). Write $$a_1,..,a_{n-1}$$ for abstract indices representing coordinates on $$\partial M$$ and $$n$$ representing the outwards normal direction. Let $$\nabla$$ be the Levi-Civita connection, and $$\partial$$ be the ordinary derivative (in local coordinates) then

\begin{align} 0 =(dw)_{na_1...a_k} & = \nabla_n w_{a_1...a_k} + \sum_i (\pm 1) \nabla_{a_i} w_{na_1...\widehat{a_i}...a_k} \\ &= \partial_n w_{a_1...a_k} + \sum_i (\pm 1) \partial_{a_i} w_{na_1...\widehat{a_i}...a_k} = \partial_n w_{a_1...a_k} \end{align} since $$w_{na_1...\widehat{a_i}...a_k} = 0$$. But I am not certain whether $$\nabla_\nu w = 0$$ or $$\partial_\nu (|w|^2)=0$$ (unless it's a flat metric).

Plugging this in, we get \begin{align} k! \ll \nabla_\nu w, |w|^{2K} w \gg_{\partial M} & =\int_{\partial M} \nabla_n w_{a_1...a_k} |w|^{2K}w^{a_1...a_k} \\ & = \sum_i (\pm 1) \int_{\partial M} \nabla_{a_i}w_{na_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} \end{align}

But the catch is that $$\nabla_{a_i}w_{na_1...\widehat{a_i}...a_k} \neq \nabla_{a_i}(\iota_\nu w)_{a_1...\widehat{a_i}...a_k}$$, so I'm stuck.

The application I have in mind uses interpolation, so $$K\in \mathbb{N}$$ was really just for simplicity's sake, and I can assume $$K$$ to be odd, $$K+1=2L$$, $$L\in \mathbb{N}$$.
Continuing from the end of my attempt: \begin{align} &\int_{\partial M} \nabla_{a_i}w_{na_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} \\ &= \int_{\partial M} \nabla_{a_i}{(\iota_\nu w)}_{a_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} - \int_{\partial M} {(\iota_{\nabla_{a_i}\nu} w)}_{a_1...\widehat{a_i}...a_k} |w|^{2K}w^{a_1...a_k} \\ \end{align} Here the first term vanishes as $$\iota_\nu w=0$$, while the second term spits out some Christoffel symbols. Since the manifold is compact, we get $$O( \int_{\partial M} |w|^{2K+2})$$.
We can control this error by the negative quantity (which I previously threw away, prematurely) $$- \ll d(|w|^2), d(|w|^{2K}) \gg_M = -K \int_M \left|d(|w|^2)\right|^2 |w|^{2(K-1)}$$ Let $$f=|w|^2$$, then we want $$C\int_{\partial M} f^{K+1} - \int_M |df|^2 f^{K-1} \lesssim \int_M f^{K+1}$$ where $$C$$ is some constant (depending on $$k,K,M$$). As $$K+1=2L$$, set $$F=f^L$$, and our problem, after simplification, becomes $$C\int_{\partial M} F^2 - \int_M |dF|^2 \lesssim \int_M F^2$$
This is just Ehrling's inequality. As $$\tau:H^1(M)\to L^2(\partial M)$$ compact and $$H^1(M)\hookrightarrow L^2(M)$$ continuous, we have $$\forall \epsilon >0, \exists C_\epsilon > 0: ||\tau F||_{L^2(\partial M)} \leq \epsilon ||F||_{H^1} + C_\epsilon ||F||_{L^2}$$