DAVOS, Switzerland—President Trump is staying in Washington to deal with the partial government shutdown. U.K. Prime Minister Theresa May is tied up with Brexit talks in London. French President Emmanuel Macron is responding to yellow-vest protests at home.

This week’s annual meeting of the World Economic Forum in this Swiss mountain resort will bring together more than 60 heads of state or government. But a number of major world leaders are skipping the gathering to attend to pressing domestic business.

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