Introduction to the theory of varifolds with applications to the min-max theory

• De Rosa, Antonio (Università Bocconi)

The construction of critical points of the area, that are not necessarily area minimizers, typically requires to pass to the limit a sequence of almost critical submanifolds with area bounds, in order to find a limit that is critical for the area. To this aim, currents are not very effective, since the mass is only lower semicontinuous and we could end up with a trivial limit. This is not an issue for the Plateau problem, because of the assigned nontrivial boundary or homology class. To solve this problem, F. J. Almgren introduced varifolds in 1965, following an earlier notion of generalized surfaces by L. C. Young in 1951. The main difference compared to currents is that the mass of varifolds is continuous in compact sets. This allows to obtain one of the most important applications of the theory of varifolds: the min-max construction of closed minimal hypersurfaces in closed Riemannian manifolds. In this mini-course, we will give an introduction to the theory of varifolds, and we will discuss how we can use varifolds to construct (possibly anisotropic) minimal hypersurfaces via the min-max theory.

Multi-bubble isoperimetric problems

• Milman, Emanuel (Euskal Herriko Unibertsitatea/Universidad del País Vasco)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems on more general spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ ($\mathbb{R}^n$ endowed with the standard Gaussian measure). Furthermore, one may consider the ``multi-bubble" isoperimetric problem, in which one prescribes the volume of $k \geq 2$ bubbles (possibly disconnected) and minimizes their total surface area -- as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $k=1$; the case $k=2$ is called the double-bubble problem, and so on. In 2000, Hutchings, Morgan, Ritor\'e and Ros resolved the double-bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently extended to $\mathbb{R}^n$) -- the standard double-bubble, whose boundary consists of three spherical caps meeting at $120^\circ$-degree angles, uniquely minimizes perimeter given prescribed volumes. A more general conjecture of J.~Sullivan from the 1990's asserts that when $k \leq n+1$, a standard $k$-bubble uniquely minimizes perimeter in $\mathbb{R}^n$. Over the last years, in collaboration with Joe Neeman, we have resolved various cases of this conjecture and its analogues in more general spaces: \\ - the multi-bubble conjecture for $k \leq n$ bubbles in Gaussian space $\mathbb{G}^n$. \\ - the multi-bubble conjecture for $k \leq \min(5,n)$ bubbles in $\mathbb{R}^n$ and $\mathbb{S}^n$, e.g. the triple-bubble conjectures when $n \geq 3$, the quadruple-bubble conjectures when $n \geq 4$, and the quintuple-bubble conjectures when $n \geq 5$ (without uniqueness on $\mathbb{R}^n$ in the latter case). In this mini course, I will present the various tools and ideas which were used to derive these results, ranging from diverse areas such as Geometric Measure Theory, Calculus of Variations, Elliptic PDE, Spectral Theory, and even a hint of Probability and Topology. I will emphasize the remaining open questions and promising directions for tackling them, which serve as fertile ground for further investigation.

Varopoulos extensions of $BMO$ and $L^p$ functions in domains with Ahlfors-regular boundaries and applications to Boundary Value problems for elliptic PDEs with $L^\infty$ coefficients.

• Mourgoglou, Mihalis (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

To be announced

• Csornyei, Marianna (University of Chicago)

Multi-bubble isoperimetric problems

• Milman, Emanuel (Euskal Herriko Unibertsitatea/Universidad del País Vasco)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems on more general spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ ($\mathbb{R}^n$ endowed with the standard Gaussian measure). Furthermore, one may consider the ``multi-bubble" isoperimetric problem, in which one prescribes the volume of $k \geq 2$ bubbles (possibly disconnected) and minimizes their total surface area -- as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $k=1$; the case $k=2$ is called the double-bubble problem, and so on. In 2000, Hutchings, Morgan, Ritor\'e and Ros resolved the double-bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently extended to $\mathbb{R}^n$) -- the standard double-bubble, whose boundary consists of three spherical caps meeting at $120^\circ$-degree angles, uniquely minimizes perimeter given prescribed volumes. A more general conjecture of J.~Sullivan from the 1990's asserts that when $k \leq n+1$, a standard $k$-bubble uniquely minimizes perimeter in $\mathbb{R}^n$. Over the last years, in collaboration with Joe Neeman, we have resolved various cases of this conjecture and its analogues in more general spaces: \\ - the multi-bubble conjecture for $k \leq n$ bubbles in Gaussian space $\mathbb{G}^n$. \\ - the multi-bubble conjecture for $k \leq \min(5,n)$ bubbles in $\mathbb{R}^n$ and $\mathbb{S}^n$, e.g. the triple-bubble conjectures when $n \geq 3$, the quadruple-bubble conjectures when $n \geq 4$, and the quintuple-bubble conjectures when $n \geq 5$ (without uniqueness on $\mathbb{R}^n$ in the latter case). In this mini course, I will present the various tools and ideas which were used to derive these results, ranging from diverse areas such as Geometric Measure Theory, Calculus of Variations, Elliptic PDE, Spectral Theory, and even a hint of Probability and Topology. I will emphasize the remaining open questions and promising directions for tackling them, which serve as fertile ground for further investigation.

Varopoulos extensions of $BMO$ and $L^p$ functions in domains with Ahlfors-regular boundaries and applications to Boundary Value problems for elliptic PDEs with $L^\infty$ coefficients.

• Mourgoglou, Mihalis (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

This course presents a unified approach to extending boundary data from rough domains into the interior, with a focus on applications to boundary value problems for elliptic operators. We study recent advances in constructing \emph{smooth harmonic-type extensions} of BMO and \( L^p \) functions from the boundary \( \partial \Omega \) of a domain \( \Omega \subset \mathbb{R}^{n+1} \), where the geometry of \( \Omega \) may be highly irregular. The domains under consideration include: \begin{itemize} \item \textbf{Corkscrew domains} when \( \partial \Omega \) is \( n \)-dimensional and Ahlfors regular, \item and \textbf{complements of \( s \)-Ahlfors regular sets} when \( s < n \). \end{itemize} The core objectives of the course include: \begin{itemize} \item Constructing \emph{smooth interior extensions} of boundary functions with optimal control in terms of \emph{Carleson measures} and \emph{non-tangential maximal functions}, \item Establishing \emph{pointwise convergence} of these extensions back to the boundary data in a non-tangential sense, \item Showing how \emph{Lipschitz boundary data} yields Lipschitz continuous extensions up to the closure of the domain. \end{itemize} A significant portion of the course will be dedicated to \textbf{applications in elliptic boundary value problems}, particularly for \emph{divergence-form elliptic systems with rough (e.g., merely bounded, complex-valued) coefficients}. We will explore: \begin{itemize} \item The role of these extensions in solving \emph{Dirichlet problems with \( L^p \) and BMO boundary data}, \item Connections between \emph{interior regularity in Carleson or tent spaces} and the \emph{solvability of Poisson problems}, \item How these tools fit into the modern framework of harmonic analysis on non-smooth domains. \end{itemize} The course is aimed at graduate students and researchers interested in \emph{elliptic PDEs, harmonic analysis, and geometric measure theory}. It will balance theoretical development with motivation from concrete problems in analysis and PDE.

Introduction to the theory of varifolds with applications to the min-max theory

• De Rosa, Antonio (Università Bocconi)

The construction of critical points of the area, that are not necessarily area minimizers, typically requires to pass to the limit a sequence of almost critical submanifolds with area bounds, in order to find a limit that is critical for the area. To this aim, currents are not very effective, since the mass is only lower semicontinuous and we could end up with a trivial limit. This is not an issue for the Plateau problem, because of the assigned nontrivial boundary or homology class. To solve this problem, F. J. Almgren introduced varifolds in 1965, following an earlier notion of generalized surfaces by L. C. Young in 1951. The main difference compared to currents is that the mass of varifolds is continuous in compact sets. This allows to obtain one of the most important applications of the theory of varifolds: the min-max construction of closed minimal hypersurfaces in closed Riemannian manifolds. In this mini-course, we will give an introduction to the theory of varifolds, and we will discuss how we can use varifolds to construct (possibly anisotropic) minimal hypersurfaces via the min-max theory.

To be announced

• Csornyei, Marianna (University of Chicago)

Varopoulos extensions of $BMO$ and $L^p$ functions in domains with Ahlfors-regular boundaries and applications to Boundary Value problems for elliptic PDEs with $L^\infty$ coefficients.

• Mourgoglou, Mihalis (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

This course presents a unified approach to extending boundary data from rough domains into the interior, with a focus on applications to boundary value problems for elliptic operators. We study recent advances in constructing \emph{smooth harmonic-type extensions} of BMO and \( L^p \) functions from the boundary \( \partial \Omega \) of a domain \( \Omega \subset \mathbb{R}^{n+1} \), where the geometry of \( \Omega \) may be highly irregular. The domains under consideration include: \begin{itemize} \item \textbf{Corkscrew domains} when \( \partial \Omega \) is \( n \)-dimensional and Ahlfors regular, \item and \textbf{complements of \( s \)-Ahlfors regular sets} when \( s < n \). \end{itemize} The core objectives of the course include: \begin{itemize} \item Constructing \emph{smooth interior extensions} of boundary functions with optimal control in terms of \emph{Carleson measures} and \emph{non-tangential maximal functions}, \item Establishing \emph{pointwise convergence} of these extensions back to the boundary data in a non-tangential sense, \item Showing how \emph{Lipschitz boundary data} yields Lipschitz continuous extensions up to the closure of the domain. \end{itemize} A significant portion of the course will be dedicated to \textbf{applications in elliptic boundary value problems}, particularly for \emph{divergence-form elliptic systems with rough (e.g., merely bounded, complex-valued) coefficients}. We will explore: \begin{itemize} \item The role of these extensions in solving \emph{Dirichlet problems with \( L^p \) and BMO boundary data}, \item Connections between \emph{interior regularity in Carleson or tent spaces} and the \emph{solvability of Poisson problems}, \item How these tools fit into the modern framework of harmonic analysis on non-smooth domains. \end{itemize} The course is aimed at graduate students and researchers interested in \emph{elliptic PDEs, harmonic analysis, and geometric measure theory}. It will balance theoretical development with motivation from concrete problems in analysis and PDE.

Introduction to the theory of varifolds with applications to the min-max theory

• De Rosa, Antonio (Università Bocconi)

The construction of critical points of the area, that are not necessarily area minimizers, typically requires to pass to the limit a sequence of almost critical submanifolds with area bounds, in order to find a limit that is critical for the area. To this aim, currents are not very effective, since the mass is only lower semicontinuous and we could end up with a trivial limit. This is not an issue for the Plateau problem, because of the assigned nontrivial boundary or homology class. To solve this problem, F. J. Almgren introduced varifolds in 1965, following an earlier notion of generalized surfaces by L. C. Young in 1951. The main difference compared to currents is that the mass of varifolds is continuous in compact sets. This allows to obtain one of the most important applications of the theory of varifolds: the min-max construction of closed minimal hypersurfaces in closed Riemannian manifolds. In this mini-course, we will give an introduction to the theory of varifolds, and we will discuss how we can use varifolds to construct (possibly anisotropic) minimal hypersurfaces via the min-max theory.

Multi-bubble isoperimetric problems

• Milman, Emanuel (Euskal Herriko Unibertsitatea/Universidad del País Vasco)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems on more general spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ ($\mathbb{R}^n$ endowed with the standard Gaussian measure). Furthermore, one may consider the ``multi-bubble" isoperimetric problem, in which one prescribes the volume of $k \geq 2$ bubbles (possibly disconnected) and minimizes their total surface area -- as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $k=1$; the case $k=2$ is called the double-bubble problem, and so on. In 2000, Hutchings, Morgan, Ritor\'e and Ros resolved the double-bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently extended to $\mathbb{R}^n$) -- the standard double-bubble, whose boundary consists of three spherical caps meeting at $120^\circ$-degree angles, uniquely minimizes perimeter given prescribed volumes. A more general conjecture of J.~Sullivan from the 1990's asserts that when $k \leq n+1$, a standard $k$-bubble uniquely minimizes perimeter in $\mathbb{R}^n$. Over the last years, in collaboration with Joe Neeman, we have resolved various cases of this conjecture and its analogues in more general spaces: \\ - the multi-bubble conjecture for $k \leq n$ bubbles in Gaussian space $\mathbb{G}^n$. \\ - the multi-bubble conjecture for $k \leq \min(5,n)$ bubbles in $\mathbb{R}^n$ and $\mathbb{S}^n$, e.g. the triple-bubble conjectures when $n \geq 3$, the quadruple-bubble conjectures when $n \geq 4$, and the quintuple-bubble conjectures when $n \geq 5$ (without uniqueness on $\mathbb{R}^n$ in the latter case). In this mini course, I will present the various tools and ideas which were used to derive these results, ranging from diverse areas such as Geometric Measure Theory, Calculus of Variations, Elliptic PDE, Spectral Theory, and even a hint of Probability and Topology. I will emphasize the remaining open questions and promising directions for tackling them, which serve as fertile ground for further investigation.

To be announced

• Csornyei, Marianna (University of Chicago)

Introduction to the theory of varifolds with applications to the min-max theory

• De Rosa, Antonio (Università Bocconi)

The construction of critical points of the area, that are not necessarily area minimizers, typically requires to pass to the limit a sequence of almost critical submanifolds with area bounds, in order to find a limit that is critical for the area. To this aim, currents are not very effective, since the mass is only lower semicontinuous and we could end up with a trivial limit. This is not an issue for the Plateau problem, because of the assigned nontrivial boundary or homology class. To solve this problem, F. J. Almgren introduced varifolds in 1965, following an earlier notion of generalized surfaces by L. C. Young in 1951. The main difference compared to currents is that the mass of varifolds is continuous in compact sets. This allows to obtain one of the most important applications of the theory of varifolds: the min-max construction of closed minimal hypersurfaces in closed Riemannian manifolds. In this mini-course, we will give an introduction to the theory of varifolds, and we will discuss how we can use varifolds to construct (possibly anisotropic) minimal hypersurfaces via the min-max theory.

Multi-bubble isoperimetric problems

• Milman, Emanuel (Euskal Herriko Unibertsitatea/Universidad del País Vasco)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems on more general spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ ($\mathbb{R}^n$ endowed with the standard Gaussian measure). Furthermore, one may consider the ``multi-bubble" isoperimetric problem, in which one prescribes the volume of $k \geq 2$ bubbles (possibly disconnected) and minimizes their total surface area -- as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $k=1$; the case $k=2$ is called the double-bubble problem, and so on. In 2000, Hutchings, Morgan, Ritor\'e and Ros resolved the double-bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently extended to $\mathbb{R}^n$) -- the standard double-bubble, whose boundary consists of three spherical caps meeting at $120^\circ$-degree angles, uniquely minimizes perimeter given prescribed volumes. A more general conjecture of J.~Sullivan from the 1990's asserts that when $k \leq n+1$, a standard $k$-bubble uniquely minimizes perimeter in $\mathbb{R}^n$. Over the last years, in collaboration with Joe Neeman, we have resolved various cases of this conjecture and its analogues in more general spaces: \\ - the multi-bubble conjecture for $k \leq n$ bubbles in Gaussian space $\mathbb{G}^n$. \\ - the multi-bubble conjecture for $k \leq \min(5,n)$ bubbles in $\mathbb{R}^n$ and $\mathbb{S}^n$, e.g. the triple-bubble conjectures when $n \geq 3$, the quadruple-bubble conjectures when $n \geq 4$, and the quintuple-bubble conjectures when $n \geq 5$ (without uniqueness on $\mathbb{R}^n$ in the latter case). In this mini course, I will present the various tools and ideas which were used to derive these results, ranging from diverse areas such as Geometric Measure Theory, Calculus of Variations, Elliptic PDE, Spectral Theory, and even a hint of Probability and Topology. I will emphasize the remaining open questions and promising directions for tackling them, which serve as fertile ground for further investigation.

Varopoulos extensions of $BMO$ and $L^p$ functions in domains with Ahlfors-regular boundaries and applications to Boundary Value problems for elliptic PDEs with $L^\infty$ coefficients.

• Mourgoglou, Mihalis (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

This course presents a unified approach to extending boundary data from rough domains into the interior, with a focus on applications to boundary value problems for elliptic operators. We study recent advances in constructing \emph{smooth harmonic-type extensions} of BMO and \( L^p \) functions from the boundary \( \partial \Omega \) of a domain \( \Omega \subset \mathbb{R}^{n+1} \), where the geometry of \( \Omega \) may be highly irregular. The domains under consideration include: \begin{itemize} \item \textbf{Corkscrew domains} when \( \partial \Omega \) is \( n \)-dimensional and Ahlfors regular, \item and \textbf{complements of \( s \)-Ahlfors regular sets} when \( s < n \). \end{itemize} The core objectives of the course include: \begin{itemize} \item Constructing \emph{smooth interior extensions} of boundary functions with optimal control in terms of \emph{Carleson measures} and \emph{non-tangential maximal functions}, \item Establishing \emph{pointwise convergence} of these extensions back to the boundary data in a non-tangential sense, \item Showing how \emph{Lipschitz boundary data} yields Lipschitz continuous extensions up to the closure of the domain. \end{itemize} A significant portion of the course will be dedicated to \textbf{applications in elliptic boundary value problems}, particularly for \emph{divergence-form elliptic systems with rough (e.g., merely bounded, complex-valued) coefficients}. We will explore: \begin{itemize} \item The role of these extensions in solving \emph{Dirichlet problems with \( L^p \) and BMO boundary data}, \item Connections between \emph{interior regularity in Carleson or tent spaces} and the \emph{solvability of Poisson problems}, \item How these tools fit into the modern framework of harmonic analysis on non-smooth domains. \end{itemize} The course is aimed at graduate students and researchers interested in \emph{elliptic PDEs, harmonic analysis, and geometric measure theory}. It will balance theoretical development with motivation from concrete problems in analysis and PDE.

To be announced

• Csornyei, Marianna (University of Chicago)